| L(s) = 1 | + 2-s + 4-s + 0.593·5-s + 8-s + 0.593·10-s + 0.593·11-s − 2.51·13-s + 16-s − 2.92·17-s + 5.38·19-s + 0.593·20-s + 0.593·22-s − 4.46·23-s − 4.64·25-s − 2.51·26-s − 6.19·29-s + 7.86·31-s + 32-s − 2.92·34-s − 37-s + 5.38·38-s + 0.593·40-s + 0.273·41-s + 11.1·43-s + 0.593·44-s − 4.46·46-s + 12.1·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.265·5-s + 0.353·8-s + 0.187·10-s + 0.178·11-s − 0.697·13-s + 0.250·16-s − 0.708·17-s + 1.23·19-s + 0.132·20-s + 0.126·22-s − 0.930·23-s − 0.929·25-s − 0.493·26-s − 1.15·29-s + 1.41·31-s + 0.176·32-s − 0.500·34-s − 0.164·37-s + 0.872·38-s + 0.0938·40-s + 0.0426·41-s + 1.70·43-s + 0.0894·44-s − 0.657·46-s + 1.77·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.370131502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.370131502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.593T + 5T^{2} \) |
| 11 | \( 1 - 0.593T + 11T^{2} \) |
| 13 | \( 1 + 2.51T + 13T^{2} \) |
| 17 | \( 1 + 2.92T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 + 4.46T + 23T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 0.273T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 8.05T + 53T^{2} \) |
| 59 | \( 1 - 8.64T + 59T^{2} \) |
| 61 | \( 1 - 6.64T + 61T^{2} \) |
| 67 | \( 1 + 1.91T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 7.91T + 73T^{2} \) |
| 79 | \( 1 + 9.24T + 79T^{2} \) |
| 83 | \( 1 + 7.70T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.57195108775269538142285455377, −7.20672211411390847042701771784, −6.29778021666941377537658206727, −5.69505258942688516612573812127, −5.12457526773139723693482644717, −4.19036564464024940490272591818, −3.72722318764231437928223296079, −2.56992013609269888269433201346, −2.10505788702511425628219423839, −0.808959057657749111890277266585,
0.808959057657749111890277266585, 2.10505788702511425628219423839, 2.56992013609269888269433201346, 3.72722318764231437928223296079, 4.19036564464024940490272591818, 5.12457526773139723693482644717, 5.69505258942688516612573812127, 6.29778021666941377537658206727, 7.20672211411390847042701771784, 7.57195108775269538142285455377
