| L(s) = 1 | − 2-s − 2.27·3-s + 4-s + 0.325·5-s + 2.27·6-s − 2.71·7-s − 8-s + 2.17·9-s − 0.325·10-s − 1.74·11-s − 2.27·12-s + 2.11·13-s + 2.71·14-s − 0.739·15-s + 16-s − 2.19·17-s − 2.17·18-s + 4.66·19-s + 0.325·20-s + 6.18·21-s + 1.74·22-s − 23-s + 2.27·24-s − 4.89·25-s − 2.11·26-s + 1.88·27-s − 2.71·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.31·3-s + 0.5·4-s + 0.145·5-s + 0.928·6-s − 1.02·7-s − 0.353·8-s + 0.724·9-s − 0.102·10-s − 0.526·11-s − 0.656·12-s + 0.586·13-s + 0.726·14-s − 0.190·15-s + 0.250·16-s − 0.531·17-s − 0.512·18-s + 1.06·19-s + 0.0727·20-s + 1.34·21-s + 0.372·22-s − 0.208·23-s + 0.464·24-s − 0.978·25-s − 0.414·26-s + 0.362·27-s − 0.513·28-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
| \(L(1)\) |
\(\approx\) |
\(0.3869677360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3869677360\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.27T + 3T^{2} \) |
| 5 | \( 1 - 0.325T + 5T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 - 2.11T + 13T^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 - 4.66T + 19T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 + 0.684T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 9.10T + 41T^{2} \) |
| 43 | \( 1 - 6.79T + 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 1.21T + 59T^{2} \) |
| 61 | \( 1 + 9.68T + 61T^{2} \) |
| 67 | \( 1 - 0.587T + 67T^{2} \) |
| 71 | \( 1 - 2.20T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 4.61T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.986801311475093909055969595334, −7.29959947209207909078605493158, −6.51863557566593990779252156504, −6.07481690031798388501149274416, −5.49134989725204518698714492331, −4.63281778173749860038837597265, −3.53585693713355979674686539712, −2.72613083301682062417428316259, −1.49948959674061008355294653961, −0.39856169041200901576338053425,
0.39856169041200901576338053425, 1.49948959674061008355294653961, 2.72613083301682062417428316259, 3.53585693713355979674686539712, 4.63281778173749860038837597265, 5.49134989725204518698714492331, 6.07481690031798388501149274416, 6.51863557566593990779252156504, 7.29959947209207909078605493158, 7.986801311475093909055969595334
