| L(s) = 1 | − 2-s − 3-s + 4-s + 4.38·5-s + 6-s + 3.17·7-s − 8-s + 9-s − 4.38·10-s − 0.317·11-s − 12-s + 13-s − 3.17·14-s − 4.38·15-s + 16-s − 0.0669·17-s − 18-s − 0.603·19-s + 4.38·20-s − 3.17·21-s + 0.317·22-s − 1.74·23-s + 24-s + 14.2·25-s − 26-s − 27-s + 3.17·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.96·5-s + 0.408·6-s + 1.20·7-s − 0.353·8-s + 0.333·9-s − 1.38·10-s − 0.0956·11-s − 0.288·12-s + 0.277·13-s − 0.849·14-s − 1.13·15-s + 0.250·16-s − 0.0162·17-s − 0.235·18-s − 0.138·19-s + 0.981·20-s − 0.693·21-s + 0.0676·22-s − 0.364·23-s + 0.204·24-s + 2.85·25-s − 0.196·26-s − 0.192·27-s + 0.600·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.275023123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.275023123\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 4.38T + 5T^{2} \) |
| 7 | \( 1 - 3.17T + 7T^{2} \) |
| 11 | \( 1 + 0.317T + 11T^{2} \) |
| 17 | \( 1 + 0.0669T + 17T^{2} \) |
| 19 | \( 1 + 0.603T + 19T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 + 3.43T + 29T^{2} \) |
| 31 | \( 1 - 8.86T + 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 + 5.16T + 41T^{2} \) |
| 43 | \( 1 + 7.69T + 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 - 1.21T + 53T^{2} \) |
| 59 | \( 1 - 3.33T + 59T^{2} \) |
| 61 | \( 1 + 1.56T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 7.48T + 71T^{2} \) |
| 73 | \( 1 - 1.78T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.103360409712559009380912526955, −6.91266366561723905163524700422, −6.50807472661907161562427977249, −5.80565041862816767129765311428, −5.16164106024411762269158530743, −4.68398445377258213210308767654, −3.29428031189360490232618927302, −2.13542627773650715616444668252, −1.77789718395072637362065327231, −0.903478823951766187924656229891,
0.903478823951766187924656229891, 1.77789718395072637362065327231, 2.13542627773650715616444668252, 3.29428031189360490232618927302, 4.68398445377258213210308767654, 5.16164106024411762269158530743, 5.80565041862816767129765311428, 6.50807472661907161562427977249, 6.91266366561723905163524700422, 8.103360409712559009380912526955
