| L(s) = 1 | + 4.43·2-s + 11.6·4-s − 9.05·5-s − 1.54·7-s + 16.2·8-s − 40.1·10-s − 7.13·11-s + 72.1·13-s − 6.87·14-s − 21.3·16-s − 44.9·17-s + 82.7·19-s − 105.·20-s − 31.6·22-s − 55.5·23-s − 42.9·25-s + 320.·26-s − 18.0·28-s − 286.·29-s + 214.·31-s − 224.·32-s − 199.·34-s + 14.0·35-s − 209.·37-s + 367.·38-s − 147.·40-s + 91.0·41-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 1.45·4-s − 0.810·5-s − 0.0836·7-s + 0.717·8-s − 1.27·10-s − 0.195·11-s + 1.54·13-s − 0.131·14-s − 0.333·16-s − 0.640·17-s + 0.999·19-s − 1.18·20-s − 0.306·22-s − 0.503·23-s − 0.343·25-s + 2.41·26-s − 0.121·28-s − 1.83·29-s + 1.24·31-s − 1.23·32-s − 1.00·34-s + 0.0677·35-s − 0.930·37-s + 1.56·38-s − 0.581·40-s + 0.346·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1899 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1899 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 211 | \( 1 + 211T \) |
| good | 2 | \( 1 - 4.43T + 8T^{2} \) |
| 5 | \( 1 + 9.05T + 125T^{2} \) |
| 7 | \( 1 + 1.54T + 343T^{2} \) |
| 11 | \( 1 + 7.13T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 55.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 286.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 214.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 209.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 91.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 97.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 272.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 503.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 626.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 520.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 293.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 877.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 694.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 751.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 380.T + 9.12e5T^{2} \) |
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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
−8.317293717450177491122371030413, −7.47900815493032120239522672790, −6.63151099721183609865454216428, −5.88315376621189499162637171256, −5.17799690386457141577057523349, −4.12197291332760897051941105850, −3.71802185540082655500582965361, −2.86414839422205271364292248259, −1.58424767933806632994756321617, 0,
1.58424767933806632994756321617, 2.86414839422205271364292248259, 3.71802185540082655500582965361, 4.12197291332760897051941105850, 5.17799690386457141577057523349, 5.88315376621189499162637171256, 6.63151099721183609865454216428, 7.47900815493032120239522672790, 8.317293717450177491122371030413
