| L(s) = 1 | − 2.20·2-s + 2.85·4-s − 1.23·5-s + 2.37·7-s − 1.88·8-s + 2.72·10-s − 1.91·11-s + 0.196·13-s − 5.22·14-s − 1.55·16-s − 1.55·17-s + 7.98·19-s − 3.53·20-s + 4.21·22-s − 3.47·25-s − 0.432·26-s + 6.77·28-s − 4.97·29-s + 1.78·31-s + 7.20·32-s + 3.43·34-s − 2.93·35-s − 3.88·37-s − 17.5·38-s + 2.33·40-s − 0.426·41-s − 4.45·43-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 1.42·4-s − 0.552·5-s + 0.896·7-s − 0.666·8-s + 0.861·10-s − 0.576·11-s + 0.0544·13-s − 1.39·14-s − 0.388·16-s − 0.378·17-s + 1.83·19-s − 0.789·20-s + 0.898·22-s − 0.694·25-s − 0.0849·26-s + 1.28·28-s − 0.924·29-s + 0.320·31-s + 1.27·32-s + 0.589·34-s − 0.495·35-s − 0.638·37-s − 2.85·38-s + 0.368·40-s − 0.0666·41-s − 0.679·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 13 | \( 1 - 0.196T + 13T^{2} \) |
| 17 | \( 1 + 1.55T + 17T^{2} \) |
| 19 | \( 1 - 7.98T + 19T^{2} \) |
| 29 | \( 1 + 4.97T + 29T^{2} \) |
| 31 | \( 1 - 1.78T + 31T^{2} \) |
| 37 | \( 1 + 3.88T + 37T^{2} \) |
| 41 | \( 1 + 0.426T + 41T^{2} \) |
| 43 | \( 1 + 4.45T + 43T^{2} \) |
| 47 | \( 1 - 2.58T + 47T^{2} \) |
| 53 | \( 1 - 9.81T + 53T^{2} \) |
| 59 | \( 1 - 7.21T + 59T^{2} \) |
| 61 | \( 1 + 7.42T + 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 - 0.730T + 71T^{2} \) |
| 73 | \( 1 + 6.44T + 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 + 4.32T + 97T^{2} \) |
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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
−8.035275905461468700626322388252, −7.41570246388649369936448697018, −7.04642689800992445033568446665, −5.79563803140454523010349636505, −5.06108756205422178254252271498, −4.15053440019130459077874013523, −3.07489851055581454424072342067, −2.02002368131735351826448776789, −1.17586547274793329740610562426, 0,
1.17586547274793329740610562426, 2.02002368131735351826448776789, 3.07489851055581454424072342067, 4.15053440019130459077874013523, 5.06108756205422178254252271498, 5.79563803140454523010349636505, 7.04642689800992445033568446665, 7.41570246388649369936448697018, 8.035275905461468700626322388252
