| L(s) = 1 | + 2.72·2-s − 3-s + 5.42·4-s − 3.51·5-s − 2.72·6-s + 7-s + 9.32·8-s + 9-s − 9.57·10-s − 0.943·11-s − 5.42·12-s − 2.69·13-s + 2.72·14-s + 3.51·15-s + 14.5·16-s + 2.72·18-s − 6.62·19-s − 19.0·20-s − 21-s − 2.56·22-s + 4.68·23-s − 9.32·24-s + 7.34·25-s − 7.33·26-s − 27-s + 5.42·28-s − 8.44·29-s + ⋯ |
| L(s) = 1 | + 1.92·2-s − 0.577·3-s + 2.71·4-s − 1.57·5-s − 1.11·6-s + 0.377·7-s + 3.29·8-s + 0.333·9-s − 3.02·10-s − 0.284·11-s − 1.56·12-s − 0.746·13-s + 0.728·14-s + 0.907·15-s + 3.64·16-s + 0.642·18-s − 1.52·19-s − 4.25·20-s − 0.218·21-s − 0.547·22-s + 0.976·23-s − 1.90·24-s + 1.46·25-s − 1.43·26-s − 0.192·27-s + 1.02·28-s − 1.56·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.72T + 2T^{2} \) |
| 5 | \( 1 + 3.51T + 5T^{2} \) |
| 11 | \( 1 + 0.943T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 19 | \( 1 + 6.62T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 + 8.44T + 29T^{2} \) |
| 31 | \( 1 + 9.03T + 31T^{2} \) |
| 37 | \( 1 - 1.45T + 37T^{2} \) |
| 41 | \( 1 - 7.57T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 3.93T + 47T^{2} \) |
| 53 | \( 1 + 6.93T + 53T^{2} \) |
| 59 | \( 1 + 3.73T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 6.72T + 67T^{2} \) |
| 71 | \( 1 + 8.39T + 71T^{2} \) |
| 73 | \( 1 - 0.788T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 5.73T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.47019977613494481579195730649, −6.89136700897609142305240428094, −6.16092919429906719091893329489, −5.19504986852068006921189645513, −4.88385645861917764766462175816, −4.04312158611968137395065474569, −3.66966867753732391769012948894, −2.66560915153251759170309286602, −1.70023337633236284852721302245, 0,
1.70023337633236284852721302245, 2.66560915153251759170309286602, 3.66966867753732391769012948894, 4.04312158611968137395065474569, 4.88385645861917764766462175816, 5.19504986852068006921189645513, 6.16092919429906719091893329489, 6.89136700897609142305240428094, 7.47019977613494481579195730649
