| L(s) = 1 | + 1.90·2-s + 1.90·3-s + 1.61·4-s + 3.61·6-s + 4.23·7-s − 0.726·8-s + 0.618·9-s − 5.85·11-s + 3.07·12-s − 3.07·13-s + 8.05·14-s − 4.61·16-s − 5.23·17-s + 1.17·18-s + 8.05·21-s − 11.1·22-s − 4.09·23-s − 1.38·24-s − 5.85·26-s − 4.53·27-s + 6.85·28-s − 2.80·29-s + 1.90·31-s − 7.33·32-s − 11.1·33-s − 9.95·34-s + 0.999·36-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 1.09·3-s + 0.809·4-s + 1.47·6-s + 1.60·7-s − 0.256·8-s + 0.206·9-s − 1.76·11-s + 0.888·12-s − 0.853·13-s + 2.15·14-s − 1.15·16-s − 1.26·17-s + 0.277·18-s + 1.75·21-s − 2.37·22-s − 0.852·23-s − 0.282·24-s − 1.14·26-s − 0.871·27-s + 1.29·28-s − 0.519·29-s + 0.341·31-s − 1.29·32-s − 1.93·33-s − 1.70·34-s + 0.166·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 3 | \( 1 - 1.90T + 3T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 11 | \( 1 + 5.85T + 11T^{2} \) |
| 13 | \( 1 + 3.07T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 31 | \( 1 - 1.90T + 31T^{2} \) |
| 37 | \( 1 - 2.80T + 37T^{2} \) |
| 41 | \( 1 + 6.88T + 41T^{2} \) |
| 43 | \( 1 + 0.381T + 43T^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 - 3.94T + 61T^{2} \) |
| 67 | \( 1 + 5.98T + 67T^{2} \) |
| 71 | \( 1 - 0.171T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + 5.25T + 79T^{2} \) |
| 83 | \( 1 - 8.76T + 83T^{2} \) |
| 89 | \( 1 - 7.77T + 89T^{2} \) |
| 97 | \( 1 - 2.62T + 97T^{2} \) |
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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
−7.61818666960757053362030214104, −6.65401389506258334832666580343, −5.66734294314587286774730887540, −5.09611144754851109528460127156, −4.65735883141102259682088182730, −3.95994281196290031332710496106, −3.03898527746209755051834103371, −2.29186905702727473181062185612, −2.04356528642148507974529970904, 0,
2.04356528642148507974529970904, 2.29186905702727473181062185612, 3.03898527746209755051834103371, 3.95994281196290031332710496106, 4.65735883141102259682088182730, 5.09611144754851109528460127156, 5.66734294314587286774730887540, 6.65401389506258334832666580343, 7.61818666960757053362030214104
