L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.64 − 0.542i)3-s − 1.00i·4-s + (2.32 − 2.32i)5-s + (−1.54 + 0.779i)6-s + (−1.76 + 1.76i)7-s + (−0.707 − 0.707i)8-s + (2.41 + 1.78i)9-s − 3.28i·10-s + (1.08 + 1.08i)11-s + (−0.542 + 1.64i)12-s + (0.766 + 3.52i)13-s + 2.49i·14-s + (−5.08 + 2.56i)15-s − 1.00·16-s − 5.73·17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.949 − 0.313i)3-s − 0.500i·4-s + (1.04 − 1.04i)5-s + (−0.631 + 0.318i)6-s + (−0.667 + 0.667i)7-s + (−0.250 − 0.250i)8-s + (0.803 + 0.594i)9-s − 1.04i·10-s + (0.327 + 0.327i)11-s + (−0.156 + 0.474i)12-s + (0.212 + 0.977i)13-s + 0.667i·14-s + (−1.31 + 0.662i)15-s − 0.250·16-s − 1.39·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.836802 - 0.562410i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.836802 - 0.562410i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.64 + 0.542i)T \) |
| 13 | \( 1 + (-0.766 - 3.52i)T \) |
good | 5 | \( 1 + (-2.32 + 2.32i)T - 5iT^{2} \) |
| 7 | \( 1 + (1.76 - 1.76i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.08 - 1.08i)T + 11iT^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + (-2.28 - 2.28i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 29 | \( 1 + 4.65iT - 29T^{2} \) |
| 31 | \( 1 + (3.82 + 3.82i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.05 - 3.05i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.410 - 0.410i)T - 41iT^{2} \) |
| 43 | \( 1 + 0.222iT - 43T^{2} \) |
| 47 | \( 1 + (7.65 + 7.65i)T + 47iT^{2} \) |
| 53 | \( 1 - 10.9iT - 53T^{2} \) |
| 59 | \( 1 + (4.65 + 4.65i)T + 59iT^{2} \) |
| 61 | \( 1 - 3.06T + 61T^{2} \) |
| 67 | \( 1 + (-0.533 - 0.533i)T + 67iT^{2} \) |
| 71 | \( 1 + (5.48 - 5.48i)T - 71iT^{2} \) |
| 73 | \( 1 + (-2.28 + 2.28i)T - 73iT^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + (-1.39 + 1.39i)T - 83iT^{2} \) |
| 89 | \( 1 + (6.41 + 6.41i)T + 89iT^{2} \) |
| 97 | \( 1 + (11.5 + 11.5i)T + 97iT^{2} \) |
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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
−13.70985641281234945737565128804, −13.05239737227398673457691208478, −12.22988529884159221146279870959, −11.26789805813860441883792601565, −9.797742668362017065530409415905, −9.023275389283640350843330541096, −6.66038575585206832989404981494, −5.71961690734794513396796625714, −4.56914739323237194620808803590, −1.85229661636046153072729210994,
3.35316169535355541291048359399, 5.19456813851508398182693589493, 6.43355932617564198985975048687, 7.00162012334420338726415072711, 9.264623025758729573648282848733, 10.49548947113355975484018925098, 11.14182548223792799006200368299, 12.83025342833026939793534517856, 13.52954049974063275935111558051, 14.68978640748632422889247770804