| L(s) = 1 | + 2-s + 3.33·3-s + 4-s + 5-s + 3.33·6-s − 0.860·7-s + 8-s + 8.10·9-s + 10-s − 1.88·11-s + 3.33·12-s + 5.65·13-s − 0.860·14-s + 3.33·15-s + 16-s − 1.67·17-s + 8.10·18-s − 7.07·19-s + 20-s − 2.86·21-s − 1.88·22-s − 3.46·23-s + 3.33·24-s + 25-s + 5.65·26-s + 17.0·27-s − 0.860·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.92·3-s + 0.5·4-s + 0.447·5-s + 1.36·6-s − 0.325·7-s + 0.353·8-s + 2.70·9-s + 0.316·10-s − 0.568·11-s + 0.961·12-s + 1.56·13-s − 0.229·14-s + 0.860·15-s + 0.250·16-s − 0.405·17-s + 1.91·18-s − 1.62·19-s + 0.223·20-s − 0.625·21-s − 0.401·22-s − 0.721·23-s + 0.680·24-s + 0.200·25-s + 1.10·26-s + 3.27·27-s − 0.162·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.425612496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.425612496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 - T \) |
| good | 3 | \( 1 - 3.33T + 3T^{2} \) |
| 7 | \( 1 + 0.860T + 7T^{2} \) |
| 11 | \( 1 + 1.88T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 - 0.0402T + 31T^{2} \) |
| 37 | \( 1 + 0.858T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 8.52T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 4.74T + 53T^{2} \) |
| 59 | \( 1 - 5.81T + 59T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 + 4.06T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 1.52T + 73T^{2} \) |
| 79 | \( 1 - 8.52T + 79T^{2} \) |
| 83 | \( 1 + 5.51T + 83T^{2} \) |
| 89 | \( 1 + 6.48T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.284520595270099330567194192239, −7.48848483416178307421797593803, −6.48392774629644625215951426765, −6.24878568349606282920270822712, −4.97154776196601879005430937808, −4.04928022735448293968629899094, −3.74530135063179183992696488676, −2.68897770231492807900738296879, −2.29910027273042545079262205153, −1.31973965220499576863148636982,
1.31973965220499576863148636982, 2.29910027273042545079262205153, 2.68897770231492807900738296879, 3.74530135063179183992696488676, 4.04928022735448293968629899094, 4.97154776196601879005430937808, 6.24878568349606282920270822712, 6.48392774629644625215951426765, 7.48848483416178307421797593803, 8.284520595270099330567194192239
