| L(s) = 1 | + 2-s + 1.61·3-s + 4-s − 5-s + 1.61·6-s + 0.618·7-s + 8-s − 0.381·9-s − 10-s + 2.85·11-s + 1.61·12-s − 7.09·13-s + 0.618·14-s − 1.61·15-s + 16-s − 6.09·17-s − 0.381·18-s − 1.85·19-s − 20-s + 1.00·21-s + 2.85·22-s + 1.61·24-s + 25-s − 7.09·26-s − 5.47·27-s + 0.618·28-s − 9.23·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.934·3-s + 0.5·4-s − 0.447·5-s + 0.660·6-s + 0.233·7-s + 0.353·8-s − 0.127·9-s − 0.316·10-s + 0.860·11-s + 0.467·12-s − 1.96·13-s + 0.165·14-s − 0.417·15-s + 0.250·16-s − 1.47·17-s − 0.0900·18-s − 0.425·19-s − 0.223·20-s + 0.218·21-s + 0.608·22-s + 0.330·24-s + 0.200·25-s − 1.39·26-s − 1.05·27-s + 0.116·28-s − 1.71·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 + 7.09T + 13T^{2} \) |
| 17 | \( 1 + 6.09T + 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 + 6.47T + 37T^{2} \) |
| 41 | \( 1 - 3.32T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 3.70T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 - 9.32T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 4.09T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.81906273187618179676538576740, −7.08648667788623873489791526150, −6.56558165991719062802854989718, −5.52344065907571606240257717444, −4.64950704676480919580323590583, −4.17903409931898013180558604652, −3.27545247173021532503507514505, −2.47777112853799117885369613054, −1.85854846336723654086047647703, 0,
1.85854846336723654086047647703, 2.47777112853799117885369613054, 3.27545247173021532503507514505, 4.17903409931898013180558604652, 4.64950704676480919580323590583, 5.52344065907571606240257717444, 6.56558165991719062802854989718, 7.08648667788623873489791526150, 7.81906273187618179676538576740
