| L(s) = 1 | − 2-s − 1.71·3-s + 4-s − 2.14·5-s + 1.71·6-s − 1.30·7-s − 8-s − 0.0419·9-s + 2.14·10-s + 4.74·11-s − 1.71·12-s + 1.19·13-s + 1.30·14-s + 3.68·15-s + 16-s − 1.31·17-s + 0.0419·18-s + 0.113·19-s − 2.14·20-s + 2.24·21-s − 4.74·22-s − 23-s + 1.71·24-s − 0.406·25-s − 1.19·26-s + 5.23·27-s − 1.30·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.992·3-s + 0.5·4-s − 0.958·5-s + 0.702·6-s − 0.493·7-s − 0.353·8-s − 0.0139·9-s + 0.677·10-s + 1.43·11-s − 0.496·12-s + 0.331·13-s + 0.349·14-s + 0.951·15-s + 0.250·16-s − 0.317·17-s + 0.00989·18-s + 0.0259·19-s − 0.479·20-s + 0.490·21-s − 1.01·22-s − 0.208·23-s + 0.351·24-s − 0.0812·25-s − 0.234·26-s + 1.00·27-s − 0.246·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 1.71T + 3T^{2} \) |
| 5 | \( 1 + 2.14T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 17 | \( 1 + 1.31T + 17T^{2} \) |
| 19 | \( 1 - 0.113T + 19T^{2} \) |
| 29 | \( 1 + 8.62T + 29T^{2} \) |
| 31 | \( 1 + 1.19T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 4.88T + 41T^{2} \) |
| 43 | \( 1 - 6.12T + 43T^{2} \) |
| 47 | \( 1 - 7.88T + 47T^{2} \) |
| 53 | \( 1 - 4.06T + 53T^{2} \) |
| 59 | \( 1 + 9.74T + 59T^{2} \) |
| 61 | \( 1 - 0.336T + 61T^{2} \) |
| 67 | \( 1 - 8.95T + 67T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 + 9.56T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 5.53T + 89T^{2} \) |
| 97 | \( 1 - 8.36T + 97T^{2} \) |
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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
−7.54744555127983959049006569834, −7.13511411838575378079200988883, −6.20998557342261472072304033866, −5.96271258449665971689218361822, −4.86855804390999418596140386076, −3.89157047286246840495638391395, −3.44248949996561541840873634296, −2.07481779835182332470161065164, −0.914356240714065580847944794014, 0,
0.914356240714065580847944794014, 2.07481779835182332470161065164, 3.44248949996561541840873634296, 3.89157047286246840495638391395, 4.86855804390999418596140386076, 5.96271258449665971689218361822, 6.20998557342261472072304033866, 7.13511411838575378079200988883, 7.54744555127983959049006569834
