| L(s) = 1 | − 2-s + 4-s − 1.06·5-s − 4.24·7-s − 8-s + 1.06·10-s − 0.241·11-s + 2.41·13-s + 4.24·14-s + 16-s − 0.230·17-s + 6.27·19-s − 1.06·20-s + 0.241·22-s + 0.998·23-s − 3.86·25-s − 2.41·26-s − 4.24·28-s − 1.94·29-s + 0.204·31-s − 32-s + 0.230·34-s + 4.51·35-s − 3.04·37-s − 6.27·38-s + 1.06·40-s + 4.79·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.476·5-s − 1.60·7-s − 0.353·8-s + 0.336·10-s − 0.0726·11-s + 0.670·13-s + 1.13·14-s + 0.250·16-s − 0.0559·17-s + 1.44·19-s − 0.238·20-s + 0.0513·22-s + 0.208·23-s − 0.773·25-s − 0.473·26-s − 0.801·28-s − 0.361·29-s + 0.0367·31-s − 0.176·32-s + 0.0395·34-s + 0.763·35-s − 0.500·37-s − 1.01·38-s + 0.168·40-s + 0.749·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7020473070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7020473070\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 0.241T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 + 0.230T + 17T^{2} \) |
| 19 | \( 1 - 6.27T + 19T^{2} \) |
| 23 | \( 1 - 0.998T + 23T^{2} \) |
| 29 | \( 1 + 1.94T + 29T^{2} \) |
| 31 | \( 1 - 0.204T + 31T^{2} \) |
| 37 | \( 1 + 3.04T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 + 6.35T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 5.75T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 8.49T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 0.674T + 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 - 0.819T + 83T^{2} \) |
| 89 | \( 1 - 3.51T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.84440806774092711892720456215, −7.16143754048132892054863701510, −6.60067938568190212919371647870, −5.93551507760839357046874525371, −5.22168897570023465791874677998, −3.99458677203351558064564045847, −3.37422946723366376108525402950, −2.82107349645398315811427209005, −1.57358768942625998052020276196, −0.46473424274636719178204327369,
0.46473424274636719178204327369, 1.57358768942625998052020276196, 2.82107349645398315811427209005, 3.37422946723366376108525402950, 3.99458677203351558064564045847, 5.22168897570023465791874677998, 5.93551507760839357046874525371, 6.60067938568190212919371647870, 7.16143754048132892054863701510, 7.84440806774092711892720456215
