| L(s) = 1 | + 6·2-s − 92·4-s − 84·5-s − 1.32e3·8-s − 504·10-s + 5.56e3·11-s + 5.15e3·13-s + 3.85e3·16-s − 1.39e4·17-s − 5.53e4·19-s + 7.72e3·20-s + 3.34e4·22-s + 9.12e4·23-s − 7.10e4·25-s + 3.09e4·26-s − 4.16e4·29-s − 1.50e5·31-s + 1.92e5·32-s − 8.39e4·34-s − 1.36e5·37-s − 3.32e5·38-s + 1.10e5·40-s − 5.10e5·41-s − 1.72e5·43-s − 5.12e5·44-s + 5.47e5·46-s − 5.19e5·47-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 0.718·4-s − 0.300·5-s − 0.911·8-s − 0.159·10-s + 1.26·11-s + 0.650·13-s + 0.235·16-s − 0.690·17-s − 1.85·19-s + 0.216·20-s + 0.668·22-s + 1.56·23-s − 0.909·25-s + 0.344·26-s − 0.316·29-s − 0.906·31-s + 1.03·32-s − 0.366·34-s − 0.442·37-s − 0.982·38-s + 0.273·40-s − 1.15·41-s − 0.330·43-s − 0.906·44-s + 0.829·46-s − 0.729·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.715638103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.715638103\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3 p T + p^{7} T^{2} \) |
| 5 | \( 1 + 84 T + p^{7} T^{2} \) |
| 11 | \( 1 - 5568 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5152 T + p^{7} T^{2} \) |
| 17 | \( 1 + 13986 T + p^{7} T^{2} \) |
| 19 | \( 1 + 55370 T + p^{7} T^{2} \) |
| 23 | \( 1 - 91272 T + p^{7} T^{2} \) |
| 29 | \( 1 + 41610 T + p^{7} T^{2} \) |
| 31 | \( 1 + 150332 T + p^{7} T^{2} \) |
| 37 | \( 1 + 136366 T + p^{7} T^{2} \) |
| 41 | \( 1 + 510258 T + p^{7} T^{2} \) |
| 43 | \( 1 + 172072 T + p^{7} T^{2} \) |
| 47 | \( 1 + 519036 T + p^{7} T^{2} \) |
| 53 | \( 1 - 59202 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1979250 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2988748 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2409404 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1504512 T + p^{7} T^{2} \) |
| 73 | \( 1 - 1821022 T + p^{7} T^{2} \) |
| 79 | \( 1 + 1669240 T + p^{7} T^{2} \) |
| 83 | \( 1 - 696738 T + p^{7} T^{2} \) |
| 89 | \( 1 - 5558490 T + p^{7} T^{2} \) |
| 97 | \( 1 + 101822 p T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.859578440224505046053937509515, −8.812376321241333436377127725658, −8.525658665363504468343471697874, −6.94727335992736406229376277004, −6.20468559586038489740764382036, −5.05314903018065627344765256378, −4.07526267205860625433608675952, −3.50467176385532323876257210982, −1.90489667370927263142017223906, −0.54628004537920749357727304440,
0.54628004537920749357727304440, 1.90489667370927263142017223906, 3.50467176385532323876257210982, 4.07526267205860625433608675952, 5.05314903018065627344765256378, 6.20468559586038489740764382036, 6.94727335992736406229376277004, 8.525658665363504468343471697874, 8.812376321241333436377127725658, 9.859578440224505046053937509515
