| L(s) = 1 | − 2-s + 4-s − 2.70·5-s + 7-s − 8-s + 2.70·10-s + 0.701·11-s + 13-s − 14-s + 16-s + 2.70·17-s − 0.701·19-s − 2.70·20-s − 0.701·22-s − 4.70·23-s + 2.29·25-s − 26-s + 28-s − 2.70·29-s − 32-s − 2.70·34-s − 2.70·35-s + 10.7·37-s + 0.701·38-s + 2.70·40-s − 3.40·41-s − 10.1·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.20·5-s + 0.377·7-s − 0.353·8-s + 0.854·10-s + 0.211·11-s + 0.277·13-s − 0.267·14-s + 0.250·16-s + 0.655·17-s − 0.160·19-s − 0.604·20-s − 0.149·22-s − 0.980·23-s + 0.459·25-s − 0.196·26-s + 0.188·28-s − 0.501·29-s − 0.176·32-s − 0.463·34-s − 0.456·35-s + 1.75·37-s + 0.113·38-s + 0.427·40-s − 0.531·41-s − 1.54·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9214473246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9214473246\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 - 0.701T + 11T^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 19 | \( 1 + 0.701T + 19T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 + 2.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 3.40T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 - 1.29T + 61T^{2} \) |
| 67 | \( 1 - 5.40T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 1.29T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 8.80T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−9.344537222605354746516440598843, −8.326669910528247763229586639342, −8.004867483443384992404400146640, −7.24407594614206697231968167030, −6.34308748696953470510688555162, −5.32836360185378698752545882362, −4.15780220342704709779066361846, −3.48011453327739545414674479268, −2.13154137431681508143391707335, −0.74237517806008437822675717051,
0.74237517806008437822675717051, 2.13154137431681508143391707335, 3.48011453327739545414674479268, 4.15780220342704709779066361846, 5.32836360185378698752545882362, 6.34308748696953470510688555162, 7.24407594614206697231968167030, 8.004867483443384992404400146640, 8.326669910528247763229586639342, 9.344537222605354746516440598843
