| L(s) = 1 | − 16·13-s + 7·16-s − 8·17-s + 16·53-s − 48·61-s − 36·81-s + 112·101-s − 72·113-s − 64·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 112·208-s + ⋯ |
| L(s) = 1 | − 4.43·13-s + 7/4·16-s − 1.94·17-s + 2.19·53-s − 6.14·61-s − 4·81-s + 11.1·101-s − 6.77·113-s − 5.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s − 7.76·208-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7108519732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7108519732\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 7 T^{4} + p^{4} T^{8} \) |
| 5 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - 94 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 802 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2702 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 2542 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 2306 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 + 5906 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 5218 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 146 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 11458 T^{4} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.41244736634111049401996388134, −5.31274471540725928290364530209, −5.22677961236133929375591998199, −4.85791199641906049639367483052, −4.63519344470472726779828732929, −4.63293773334837399421387742549, −4.60151023960271146503900323452, −4.38704614279648199788569428496, −4.38676027525123866439003553849, −4.05977718088841325333421298362, −3.82791747069429316225903917768, −3.64443011260866052772093449721, −3.54996255205624054556078189226, −3.06272147379643403282507543331, −2.93865623041988867122046125664, −2.91026128516531628028517714482, −2.84655685631853651544109379918, −2.60328408124032007000203627933, −2.23800383597365298541901517961, −2.17551574265596023016345934681, −1.82956780279165619432200251846, −1.59227403407412329797455187407, −1.54820177126435894202870823023, −0.71171634099338401619364074715, −0.30605336804112161333234131524,
0.30605336804112161333234131524, 0.71171634099338401619364074715, 1.54820177126435894202870823023, 1.59227403407412329797455187407, 1.82956780279165619432200251846, 2.17551574265596023016345934681, 2.23800383597365298541901517961, 2.60328408124032007000203627933, 2.84655685631853651544109379918, 2.91026128516531628028517714482, 2.93865623041988867122046125664, 3.06272147379643403282507543331, 3.54996255205624054556078189226, 3.64443011260866052772093449721, 3.82791747069429316225903917768, 4.05977718088841325333421298362, 4.38676027525123866439003553849, 4.38704614279648199788569428496, 4.60151023960271146503900323452, 4.63293773334837399421387742549, 4.63519344470472726779828732929, 4.85791199641906049639367483052, 5.22677961236133929375591998199, 5.31274471540725928290364530209, 5.41244736634111049401996388134
Plot not available for L-functions of degree greater than 10.
