| L(s) = 1 | + 4.48·2-s − 3.10·3-s + 12.1·4-s + 9.82·5-s − 13.9·6-s + 8.62·7-s + 18.5·8-s − 17.3·9-s + 44.0·10-s − 11·11-s − 37.6·12-s + 38.6·14-s − 30.4·15-s − 13.9·16-s − 17.7·17-s − 77.9·18-s − 11.7·19-s + 119.·20-s − 26.7·21-s − 49.3·22-s − 22.8·23-s − 57.3·24-s − 28.4·25-s + 137.·27-s + 104.·28-s − 239.·29-s − 136.·30-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.596·3-s + 1.51·4-s + 0.878·5-s − 0.946·6-s + 0.465·7-s + 0.817·8-s − 0.643·9-s + 1.39·10-s − 0.301·11-s − 0.904·12-s + 0.738·14-s − 0.524·15-s − 0.218·16-s − 0.253·17-s − 1.02·18-s − 0.142·19-s + 1.33·20-s − 0.277·21-s − 0.478·22-s − 0.206·23-s − 0.488·24-s − 0.227·25-s + 0.981·27-s + 0.705·28-s − 1.53·29-s − 0.832·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.48T + 8T^{2} \) |
| 3 | \( 1 + 3.10T + 27T^{2} \) |
| 5 | \( 1 - 9.82T + 125T^{2} \) |
| 7 | \( 1 - 8.62T + 343T^{2} \) |
| 17 | \( 1 + 17.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 11.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 22.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 239.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 52.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 146.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 84.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 102.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 267.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 36.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 557.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 324.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 243.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 667.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 41.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 2.12T + 7.04e5T^{2} \) |
| 97 | \( 1 + 188.T + 9.12e5T^{2} \) |
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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
−8.451146265154941603424121317664, −7.32990453353161789763898784466, −6.47945050184857442265999463149, −5.65462660057718083986309208538, −5.46643286269077042490113164168, −4.55032849423456335774152391831, −3.59506985115793917678465635345, −2.54886995025307509339813656340, −1.74589696948279018004660645033, 0,
1.74589696948279018004660645033, 2.54886995025307509339813656340, 3.59506985115793917678465635345, 4.55032849423456335774152391831, 5.46643286269077042490113164168, 5.65462660057718083986309208538, 6.47945050184857442265999463149, 7.32990453353161789763898784466, 8.451146265154941603424121317664
