| L(s) = 1 | − 3.87·2-s + 6.98·4-s + 17.8·5-s + 4.77·7-s + 3.93·8-s − 69.0·10-s − 1.33·11-s + 66.8·13-s − 18.4·14-s − 71.0·16-s − 126.·17-s + 41.7·19-s + 124.·20-s + 5.15·22-s + 43.9·23-s + 193.·25-s − 258.·26-s + 33.3·28-s − 191.·29-s + 83.8·31-s + 243.·32-s + 490.·34-s + 85.2·35-s − 4.03·37-s − 161.·38-s + 70.1·40-s − 55.8·41-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 0.873·4-s + 1.59·5-s + 0.257·7-s + 0.173·8-s − 2.18·10-s − 0.0365·11-s + 1.42·13-s − 0.352·14-s − 1.11·16-s − 1.80·17-s + 0.503·19-s + 1.39·20-s + 0.0499·22-s + 0.398·23-s + 1.54·25-s − 1.95·26-s + 0.225·28-s − 1.22·29-s + 0.485·31-s + 1.34·32-s + 2.47·34-s + 0.411·35-s − 0.0179·37-s − 0.689·38-s + 0.277·40-s − 0.212·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 3.87T + 8T^{2} \) |
| 5 | \( 1 - 17.8T + 125T^{2} \) |
| 7 | \( 1 - 4.77T + 343T^{2} \) |
| 11 | \( 1 + 1.33T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 41.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 43.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 83.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 4.03T + 5.06e4T^{2} \) |
| 41 | \( 1 + 55.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 79.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 590.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 253.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 429.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 305.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 304.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 421.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 385.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 112.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 110.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 809.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 80.5T + 9.12e5T^{2} \) |
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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
−8.617840220313548025466525241581, −7.82011202894961344034262374636, −6.70771526248291913511389954211, −6.31345623202443872912279227607, −5.30300319703151597051379218831, −4.37580987961735660933808302011, −2.93199791937135795693398915163, −1.76408581454575104363528245042, −1.43438352243515317254026478227, 0,
1.43438352243515317254026478227, 1.76408581454575104363528245042, 2.93199791937135795693398915163, 4.37580987961735660933808302011, 5.30300319703151597051379218831, 6.31345623202443872912279227607, 6.70771526248291913511389954211, 7.82011202894961344034262374636, 8.617840220313548025466525241581
