| L(s) = 1 | + 5.28·2-s + 0.753·3-s + 19.9·4-s − 2.74·5-s + 3.98·6-s + 63.1·8-s − 26.4·9-s − 14.5·10-s − 22.2·11-s + 15.0·12-s + 83.5·13-s − 2.07·15-s + 174.·16-s − 59.0·17-s − 139.·18-s + 121.·19-s − 54.7·20-s − 117.·22-s + 148.·23-s + 47.5·24-s − 117.·25-s + 441.·26-s − 40.2·27-s + 117.·29-s − 10.9·30-s − 39.1·31-s + 415.·32-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 0.145·3-s + 2.49·4-s − 0.245·5-s + 0.271·6-s + 2.78·8-s − 0.978·9-s − 0.459·10-s − 0.611·11-s + 0.361·12-s + 1.78·13-s − 0.0356·15-s + 2.71·16-s − 0.843·17-s − 1.82·18-s + 1.46·19-s − 0.612·20-s − 1.14·22-s + 1.34·23-s + 0.404·24-s − 0.939·25-s + 3.33·26-s − 0.287·27-s + 0.749·29-s − 0.0665·30-s − 0.226·31-s + 2.29·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.571332758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.571332758\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 - 5.28T + 8T^{2} \) |
| 3 | \( 1 - 0.753T + 27T^{2} \) |
| 5 | \( 1 + 2.74T + 125T^{2} \) |
| 11 | \( 1 + 22.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 83.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 121.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 148.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 117.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 39.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 8.48T + 5.06e4T^{2} \) |
| 41 | \( 1 - 411.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 264.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 549.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 814.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 484.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 533.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 619.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 978.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 162.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 328.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 363.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−8.446849579586637709798518372650, −7.65840959450206658933684421348, −6.82610675296105153722665116184, −5.95586580591274710245289694787, −5.53747019310914652884718332173, −4.64234767452381213177466206898, −3.73936063190186096591462640296, −3.11503464548469005598700398862, −2.35396899588445075408663125274, −1.00775846966406850999452178189,
1.00775846966406850999452178189, 2.35396899588445075408663125274, 3.11503464548469005598700398862, 3.73936063190186096591462640296, 4.64234767452381213177466206898, 5.53747019310914652884718332173, 5.95586580591274710245289694787, 6.82610675296105153722665116184, 7.65840959450206658933684421348, 8.446849579586637709798518372650
