| L(s) = 1 | − i·5-s + 4i·7-s − 3·9-s + 4i·11-s − 3·17-s − 4·23-s + 4·25-s − 29-s − 4i·31-s + 4·35-s − 3i·37-s − 9i·41-s − 8·43-s + 3i·45-s − 8i·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 1.51i·7-s − 9-s + 1.20i·11-s − 0.727·17-s − 0.834·23-s + 0.800·25-s − 0.185·29-s − 0.718i·31-s + 0.676·35-s − 0.493i·37-s − 1.40i·41-s − 1.21·43-s + 0.447i·45-s − 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + iT - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 + 9iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8iT - 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.610769824295248471041951273504, −8.001072712767556716062830166160, −6.92578603339205590747498603287, −6.13204976124694715392804274147, −5.35572706140883841840667249209, −4.81823495389662608287982543811, −3.67480412635643553567613405086, −2.46951662513606843658717809492, −1.96436314066399732622078199988, 0,
1.24757403827324554137466200117, 2.79695592966418067312060783839, 3.42345358205253223299657707854, 4.33328282280001880555721790039, 5.26297422473096205932181692944, 6.34327232334369413726302767679, 6.67025700886463058125137020993, 7.74324969950954866391423347703, 8.286889706379832922663640540182
