| L(s) = 1 | + 1.63·2-s + 0.857·3-s + 0.670·4-s + 5-s + 1.40·6-s − 2.17·8-s − 2.26·9-s + 1.63·10-s + 4.94·11-s + 0.574·12-s − 0.368·13-s + 0.857·15-s − 4.89·16-s − 0.188·17-s − 3.70·18-s − 5.40·19-s + 0.670·20-s + 8.08·22-s − 4.35·23-s − 1.86·24-s + 25-s − 0.602·26-s − 4.51·27-s − 5.36·29-s + 1.40·30-s − 31-s − 3.64·32-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.495·3-s + 0.335·4-s + 0.447·5-s + 0.572·6-s − 0.768·8-s − 0.754·9-s + 0.516·10-s + 1.49·11-s + 0.165·12-s − 0.102·13-s + 0.221·15-s − 1.22·16-s − 0.0456·17-s − 0.872·18-s − 1.24·19-s + 0.149·20-s + 1.72·22-s − 0.908·23-s − 0.380·24-s + 0.200·25-s − 0.118·26-s − 0.868·27-s − 0.996·29-s + 0.255·30-s − 0.179·31-s − 0.644·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.63T + 2T^{2} \) |
| 3 | \( 1 - 0.857T + 3T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 13 | \( 1 + 0.368T + 13T^{2} \) |
| 17 | \( 1 + 0.188T + 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 23 | \( 1 + 4.35T + 23T^{2} \) |
| 29 | \( 1 + 5.36T + 29T^{2} \) |
| 37 | \( 1 + 1.96T + 37T^{2} \) |
| 41 | \( 1 - 2.71T + 41T^{2} \) |
| 43 | \( 1 - 2.74T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + 0.237T + 53T^{2} \) |
| 59 | \( 1 - 1.36T + 59T^{2} \) |
| 61 | \( 1 - 0.536T + 61T^{2} \) |
| 67 | \( 1 + 8.34T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 3.26T + 73T^{2} \) |
| 79 | \( 1 - 0.172T + 79T^{2} \) |
| 83 | \( 1 - 3.05T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 9.39T + 97T^{2} \) |
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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
−7.39174348232269220975798939001, −6.47859686936308341142310785481, −6.07644036993535289370708345627, −5.48347687853997419346953056766, −4.51048769951876261267797661899, −3.96287526029141309163133405431, −3.32680610683708419141403410717, −2.44889610941162397359812222666, −1.67834282779504556632525932574, 0,
1.67834282779504556632525932574, 2.44889610941162397359812222666, 3.32680610683708419141403410717, 3.96287526029141309163133405431, 4.51048769951876261267797661899, 5.48347687853997419346953056766, 6.07644036993535289370708345627, 6.47859686936308341142310785481, 7.39174348232269220975798939001
