| L(s) = 1 | + 2-s − 1.78·3-s + 4-s − 1.18·5-s − 1.78·6-s − 7-s + 8-s + 0.191·9-s − 1.18·10-s − 2.39·11-s − 1.78·12-s − 0.479·13-s − 14-s + 2.11·15-s + 16-s + 1.12·17-s + 0.191·18-s − 1.80·19-s − 1.18·20-s + 1.78·21-s − 2.39·22-s − 8.54·23-s − 1.78·24-s − 3.59·25-s − 0.479·26-s + 5.01·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.03·3-s + 0.5·4-s − 0.529·5-s − 0.729·6-s − 0.377·7-s + 0.353·8-s + 0.0639·9-s − 0.374·10-s − 0.722·11-s − 0.515·12-s − 0.133·13-s − 0.267·14-s + 0.546·15-s + 0.250·16-s + 0.273·17-s + 0.0452·18-s − 0.413·19-s − 0.264·20-s + 0.389·21-s − 0.510·22-s − 1.78·23-s − 0.364·24-s − 0.719·25-s − 0.0940·26-s + 0.965·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9878760216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9878760216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 11 | \( 1 + 2.39T + 11T^{2} \) |
| 13 | \( 1 + 0.479T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 23 | \( 1 + 8.54T + 23T^{2} \) |
| 29 | \( 1 - 9.94T + 29T^{2} \) |
| 31 | \( 1 - 3.16T + 31T^{2} \) |
| 37 | \( 1 - 0.0228T + 37T^{2} \) |
| 41 | \( 1 + 8.29T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 1.10T + 47T^{2} \) |
| 53 | \( 1 - 0.735T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 1.57T + 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 - 1.64T + 73T^{2} \) |
| 79 | \( 1 - 3.54T + 79T^{2} \) |
| 83 | \( 1 - 4.65T + 83T^{2} \) |
| 89 | \( 1 - 1.46T + 89T^{2} \) |
| 97 | \( 1 - 7.81T + 97T^{2} \) |
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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
−8.116433689978209260377897857534, −7.15838536173246634270115118133, −6.31278743892222705483571365735, −6.08137679260814494175558990620, −5.08903796482827411848299166700, −4.67495245652032455472041267269, −3.72636085191561925638433486656, −2.96617331295594383818657603960, −1.95256734522458068809698245161, −0.47029735449431399225886410545,
0.47029735449431399225886410545, 1.95256734522458068809698245161, 2.96617331295594383818657603960, 3.72636085191561925638433486656, 4.67495245652032455472041267269, 5.08903796482827411848299166700, 6.08137679260814494175558990620, 6.31278743892222705483571365735, 7.15838536173246634270115118133, 8.116433689978209260377897857534
