| L(s) = 1 | + 4.46·2-s − 3·3-s + 11.9·4-s − 10.8·5-s − 13.3·6-s + 19.5·7-s + 17.6·8-s + 9·9-s − 48.5·10-s − 11·11-s − 35.8·12-s + 76.7·13-s + 87.2·14-s + 32.5·15-s − 16.8·16-s − 67.6·17-s + 40.1·18-s − 131.·19-s − 129.·20-s − 58.6·21-s − 49.1·22-s − 7.91·23-s − 52.8·24-s − 7.03·25-s + 342.·26-s − 27·27-s + 233.·28-s + ⋯ |
| L(s) = 1 | + 1.57·2-s − 0.577·3-s + 1.49·4-s − 0.971·5-s − 0.911·6-s + 1.05·7-s + 0.779·8-s + 0.333·9-s − 1.53·10-s − 0.301·11-s − 0.862·12-s + 1.63·13-s + 1.66·14-s + 0.560·15-s − 0.263·16-s − 0.965·17-s + 0.526·18-s − 1.58·19-s − 1.45·20-s − 0.608·21-s − 0.476·22-s − 0.0717·23-s − 0.449·24-s − 0.0562·25-s + 2.58·26-s − 0.192·27-s + 1.57·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 4.46T + 8T^{2} \) |
| 5 | \( 1 + 10.8T + 125T^{2} \) |
| 7 | \( 1 - 19.5T + 343T^{2} \) |
| 13 | \( 1 - 76.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 67.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.91T + 1.21e4T^{2} \) |
| 29 | \( 1 - 293.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 178.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 49.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 129.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 262.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 68.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 636.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 846.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 863.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 388.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 605.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 283.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 339.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 120.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3T + 9.12e5T^{2} \) |
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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
−8.354432156406336969935678917919, −7.38305471131813307139845851286, −6.47132139806103607574247471646, −5.94190447837430456710204464788, −4.95908215485821609484353707568, −4.25283904810584956161518811360, −3.92186970124121933498610953482, −2.64595638396143322697638838427, −1.49795826374016014095172586061, 0,
1.49795826374016014095172586061, 2.64595638396143322697638838427, 3.92186970124121933498610953482, 4.25283904810584956161518811360, 4.95908215485821609484353707568, 5.94190447837430456710204464788, 6.47132139806103607574247471646, 7.38305471131813307139845851286, 8.354432156406336969935678917919
