| L(s) = 1 | + 10.5·2-s − 10.8·3-s + 78.5·4-s − 114.·6-s − 153.·7-s + 489.·8-s − 125.·9-s + 87.2·11-s − 853.·12-s − 901.·13-s − 1.61e3·14-s + 2.63e3·16-s + 193.·17-s − 1.31e3·18-s + 2.51e3·19-s + 1.66e3·21-s + 917.·22-s − 741.·23-s − 5.31e3·24-s − 9.48e3·26-s + 3.99e3·27-s − 1.20e4·28-s + 1.39e3·29-s + 1.39e3·31-s + 1.20e4·32-s − 947.·33-s + 2.03e3·34-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 0.696·3-s + 2.45·4-s − 1.29·6-s − 1.18·7-s + 2.70·8-s − 0.514·9-s + 0.217·11-s − 1.71·12-s − 1.47·13-s − 2.19·14-s + 2.57·16-s + 0.162·17-s − 0.956·18-s + 1.59·19-s + 0.824·21-s + 0.403·22-s − 0.292·23-s − 1.88·24-s − 2.75·26-s + 1.05·27-s − 2.90·28-s + 0.308·29-s + 0.260·31-s + 2.07·32-s − 0.151·33-s + 0.302·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.686801300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.686801300\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 - 10.5T + 32T^{2} \) |
| 3 | \( 1 + 10.8T + 243T^{2} \) |
| 7 | \( 1 + 153.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 87.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + 901.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 193.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.51e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 741.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.56e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.03e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 2.03e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.14e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.56e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.45e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.81e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.11e4T + 8.58e9T^{2} \) |
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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
−9.512964518533994975671237850057, −7.905700076854556325545059685399, −6.88669239995996808283544640813, −6.48230156384248850588161478850, −5.41468145561405914228404063247, −5.15277266481209525851129680425, −3.93552919555868722773715113754, −3.07488696590271914877024333893, −2.38330339227751347181705268471, −0.69336562585408233613511636235,
0.69336562585408233613511636235, 2.38330339227751347181705268471, 3.07488696590271914877024333893, 3.93552919555868722773715113754, 5.15277266481209525851129680425, 5.41468145561405914228404063247, 6.48230156384248850588161478850, 6.88669239995996808283544640813, 7.905700076854556325545059685399, 9.512964518533994975671237850057
