| L(s) = 1 | − 8·4-s + 32·9-s + 16·11-s + 36·16-s − 16·23-s + 8·25-s − 256·36-s − 80·37-s − 128·44-s + 32·53-s − 120·64-s + 16·67-s − 48·71-s + 512·81-s + 128·92-s + 512·99-s − 64·100-s + 96·113-s + 124·121-s + 127-s + 131-s + 137-s + 139-s + 1.15e3·144-s + 640·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4·4-s + 32/3·9-s + 4.82·11-s + 9·16-s − 3.33·23-s + 8/5·25-s − 42.6·36-s − 13.1·37-s − 19.2·44-s + 4.39·53-s − 15·64-s + 1.95·67-s − 5.69·71-s + 56.8·81-s + 13.3·92-s + 51.4·99-s − 6.39·100-s + 9.03·113-s + 11.2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 96·144-s + 52.6·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(31.21444459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(31.21444459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 7 | \( 1 \) |
| 11 | \( ( 1 - 8 T + 34 T^{2} - 128 T^{3} + 450 T^{4} - 128 p T^{5} + 34 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| good | 3 | \( ( 1 - 8 T^{2} + 32 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 5 | \( ( 1 - 4 T^{2} + 14 p T^{4} - 196 T^{6} + 2306 T^{8} - 196 p^{2} T^{10} + 14 p^{5} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 + 16 T^{2} + 84 T^{4} + 2512 T^{6} + 70838 T^{8} + 2512 p^{2} T^{10} + 84 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 - 12 T^{2} + 486 T^{4} - 9772 T^{6} + 117954 T^{8} - 9772 p^{2} T^{10} + 486 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 + 60 T^{2} + 1806 T^{4} + 48748 T^{6} + 1112226 T^{8} + 48748 p^{2} T^{10} + 1806 p^{4} T^{12} + 60 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 + 4 T + 54 T^{2} + 84 T^{3} + 1282 T^{4} + 84 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 48 T^{2} + 2300 T^{4} - 86352 T^{6} + 2398374 T^{8} - 86352 p^{2} T^{10} + 2300 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 - 116 T^{2} + 7990 T^{4} - 378596 T^{6} + 13503266 T^{8} - 378596 p^{2} T^{10} + 7990 p^{4} T^{12} - 116 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 + 20 T + 216 T^{2} + 1644 T^{3} + 10654 T^{4} + 1644 p T^{5} + 216 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 260 T^{2} + 31398 T^{4} + 2313572 T^{6} + 114345410 T^{8} + 2313572 p^{2} T^{10} + 31398 p^{4} T^{12} + 260 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 212 T^{2} + 23336 T^{4} - 1672476 T^{6} + 84829678 T^{8} - 1672476 p^{2} T^{10} + 23336 p^{4} T^{12} - 212 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 - 324 T^{2} + 47862 T^{4} - 4216212 T^{6} + 242387618 T^{8} - 4216212 p^{2} T^{10} + 47862 p^{4} T^{12} - 324 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 8 T + 202 T^{2} - 1136 T^{3} + 15738 T^{4} - 1136 p T^{5} + 202 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 - 352 T^{2} + 58432 T^{4} - 6022240 T^{6} + 424194146 T^{8} - 6022240 p^{2} T^{10} + 58432 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 + 64 T^{2} + 14676 T^{4} + 673312 T^{6} + 81782582 T^{8} + 673312 p^{2} T^{10} + 14676 p^{4} T^{12} + 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 4 T + 214 T^{2} - 508 T^{3} + 19322 T^{4} - 508 p T^{5} + 214 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 + 12 T + 222 T^{2} + 1644 T^{3} + 19874 T^{4} + 1644 p T^{5} + 222 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 + 228 T^{2} + 35526 T^{4} + 3992068 T^{6} + 324806274 T^{8} + 3992068 p^{2} T^{10} + 35526 p^{4} T^{12} + 228 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{8} \) |
| 83 | \( ( 1 + 348 T^{2} + 67182 T^{4} + 8774924 T^{6} + 846244962 T^{8} + 8774924 p^{2} T^{10} + 67182 p^{4} T^{12} + 348 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 - 480 T^{2} + 110400 T^{4} - 16169952 T^{6} + 1681034690 T^{8} - 16169952 p^{2} T^{10} + 110400 p^{4} T^{12} - 480 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 288 T^{2} + 38304 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.41050696040170598134096126531, −2.35612711403107512679153491749, −2.14338868874303598935837224101, −2.13819699158310587612649567526, −2.07942385577589925890882262311, −1.94426881576590211043873490794, −1.88499082017537163409391836131, −1.86690043279335628738825835517, −1.85755210577548172187197347877, −1.65499261891564630094230825829, −1.63976039107274546843277801049, −1.57496718668128786031841974846, −1.57225787955876252848544432654, −1.53783445611018597141774004930, −1.37833599692662527291451119480, −1.34396196380207522792031512907, −1.31380076510262788926795596808, −0.970006485809199657894570528139, −0.937319888817910696572595836168, −0.929652384232676273942279992610, −0.800223549264476022611171370637, −0.78862876064001558754242661617, −0.47575819445438652828113327248, −0.31267066194336955763433577780, −0.23227231093499160024724149523,
0.23227231093499160024724149523, 0.31267066194336955763433577780, 0.47575819445438652828113327248, 0.78862876064001558754242661617, 0.800223549264476022611171370637, 0.929652384232676273942279992610, 0.937319888817910696572595836168, 0.970006485809199657894570528139, 1.31380076510262788926795596808, 1.34396196380207522792031512907, 1.37833599692662527291451119480, 1.53783445611018597141774004930, 1.57225787955876252848544432654, 1.57496718668128786031841974846, 1.63976039107274546843277801049, 1.65499261891564630094230825829, 1.85755210577548172187197347877, 1.86690043279335628738825835517, 1.88499082017537163409391836131, 1.94426881576590211043873490794, 2.07942385577589925890882262311, 2.13819699158310587612649567526, 2.14338868874303598935837224101, 2.35612711403107512679153491749, 2.41050696040170598134096126531
Plot not available for L-functions of degree greater than 10.
