| L(s) = 1 | − 1.17·2-s + 0.381·4-s − 7-s + 0.726·8-s + 9-s + 1.61·11-s + 1.17·14-s − 1.23·16-s − 1.17·18-s − 1.90·22-s + 0.618·23-s + 25-s − 0.381·28-s − 1.90·29-s + 0.726·32-s + 0.381·36-s − 1.61·43-s + 0.618·44-s − 0.726·46-s + 49-s − 1.17·50-s + 1.17·53-s − 0.726·56-s + 2.23·58-s − 63-s + 0.381·64-s + 1.17·67-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.381·4-s − 7-s + 0.726·8-s + 9-s + 1.61·11-s + 1.17·14-s − 1.23·16-s − 1.17·18-s − 1.90·22-s + 0.618·23-s + 25-s − 0.381·28-s − 1.90·29-s + 0.726·32-s + 0.381·36-s − 1.61·43-s + 0.618·44-s − 0.726·46-s + 49-s − 1.17·50-s + 1.17·53-s − 0.726·56-s + 2.23·58-s − 63-s + 0.381·64-s + 1.17·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6629253917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6629253917\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.17T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.61T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 + 1.90T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.17T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.17T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.90T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.203743631023791004363333834341, −8.621624232741404911220456351245, −7.54079475970765626826531841508, −6.89879094495389046916886732604, −6.47995354696416956065557806463, −5.17687471776181476460598889551, −4.12891555653162599963939373919, −3.48344006436650449533195192636, −1.94796406838159161512101365954, −0.970010285617325072201697358933,
0.970010285617325072201697358933, 1.94796406838159161512101365954, 3.48344006436650449533195192636, 4.12891555653162599963939373919, 5.17687471776181476460598889551, 6.47995354696416956065557806463, 6.89879094495389046916886732604, 7.54079475970765626826531841508, 8.621624232741404911220456351245, 9.203743631023791004363333834341
