| L(s) = 1 | − 4·4-s + 4·7-s + 12·13-s + 10·16-s + 8·25-s − 16·28-s − 4·37-s − 28·43-s + 18·49-s − 48·52-s − 20·64-s + 40·67-s − 72·73-s + 40·79-s + 48·91-s + 12·97-s − 32·100-s − 60·103-s + 20·109-s + 40·112-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 16·148-s + 149-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.51·7-s + 3.32·13-s + 5/2·16-s + 8/5·25-s − 3.02·28-s − 0.657·37-s − 4.26·43-s + 18/7·49-s − 6.65·52-s − 5/2·64-s + 4.88·67-s − 8.42·73-s + 4.50·79-s + 5.03·91-s + 1.21·97-s − 3.19·100-s − 5.91·103-s + 1.91·109-s + 3.77·112-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6893252152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6893252152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} - 105 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 6 T + 35 T^{2} - 138 T^{3} + 516 T^{4} - 138 p T^{5} + 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 28 T^{2} + 495 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 14 T^{2} - 165 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 + 8 T^{2} + 958 T^{4} - 20608 T^{6} + 38131 T^{8} - 20608 p^{2} T^{10} + 958 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 10 T^{2} + 1299 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T - 53 T^{2} - 34 T^{3} + 1732 T^{4} - 34 p T^{5} - 53 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 76 T^{2} + 4095 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 40 T^{2} + 2226 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 4 T^{2} + 4762 T^{4} + 41456 T^{6} + 14589091 T^{8} + 41456 p^{2} T^{10} + 4762 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 226 T^{2} + 20139 T^{4} - 226 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 10 T + 87 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 79107 T^{4} + 8280 p T^{5} + 662 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 10 T + 165 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 - 68 T^{2} + 58 T^{4} + 626416 T^{6} - 58464701 T^{8} + 626416 p^{2} T^{10} + 58 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 32 T^{2} + 8254 T^{4} + 738304 T^{6} - 18580445 T^{8} + 738304 p^{2} T^{10} + 8254 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 6 T + 185 T^{2} - 1038 T^{3} + 21684 T^{4} - 1038 p T^{5} + 185 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.18340892436498822396267130306, −3.93692912924853673145816888303, −3.90822630776541845787388495475, −3.85089260871354659326795005598, −3.69104136712967956785736267832, −3.66599582916346057857494838823, −3.59216824933628714410565689020, −3.33826760924467804447313655104, −3.14513270788645663559158148829, −2.99814123435752513156012241553, −2.90007737509954269073228635719, −2.80912391257787855643239534409, −2.76902668104428103864772485950, −2.33451403757104796606720910326, −2.11064634769194501961065299653, −1.97792685605643215714491683218, −1.86454365711732429432244125663, −1.77526441997128476014229279371, −1.45378501618859383646333811994, −1.22544189118304985333708937457, −1.19693929740338588308961457731, −1.04212164399707012814132861656, −0.947834028491759467749707402237, −0.51718394439874704471367590391, −0.098847394691252180090641543823,
0.098847394691252180090641543823, 0.51718394439874704471367590391, 0.947834028491759467749707402237, 1.04212164399707012814132861656, 1.19693929740338588308961457731, 1.22544189118304985333708937457, 1.45378501618859383646333811994, 1.77526441997128476014229279371, 1.86454365711732429432244125663, 1.97792685605643215714491683218, 2.11064634769194501961065299653, 2.33451403757104796606720910326, 2.76902668104428103864772485950, 2.80912391257787855643239534409, 2.90007737509954269073228635719, 2.99814123435752513156012241553, 3.14513270788645663559158148829, 3.33826760924467804447313655104, 3.59216824933628714410565689020, 3.66599582916346057857494838823, 3.69104136712967956785736267832, 3.85089260871354659326795005598, 3.90822630776541845787388495475, 3.93692912924853673145816888303, 4.18340892436498822396267130306
Plot not available for L-functions of degree greater than 10.
