| L(s) = 1 | − 2-s − 0.302·3-s + 4-s + 3.48·5-s + 0.302·6-s + 7-s − 8-s − 2.90·9-s − 3.48·10-s − 1.91·11-s − 0.302·12-s − 1.92·13-s − 14-s − 1.05·15-s + 16-s + 1.82·17-s + 2.90·18-s + 1.01·19-s + 3.48·20-s − 0.302·21-s + 1.91·22-s + 6.54·23-s + 0.302·24-s + 7.12·25-s + 1.92·26-s + 1.78·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.174·3-s + 0.5·4-s + 1.55·5-s + 0.123·6-s + 0.377·7-s − 0.353·8-s − 0.969·9-s − 1.10·10-s − 0.577·11-s − 0.0874·12-s − 0.533·13-s − 0.267·14-s − 0.272·15-s + 0.250·16-s + 0.442·17-s + 0.685·18-s + 0.233·19-s + 0.778·20-s − 0.0661·21-s + 0.408·22-s + 1.36·23-s + 0.0618·24-s + 1.42·25-s + 0.377·26-s + 0.344·27-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
| \(L(1)\) |
\(\approx\) |
\(1.681799603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.681799603\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.302T + 3T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 13 | \( 1 + 1.92T + 13T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 - 6.54T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 - 0.511T + 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 + 0.862T + 41T^{2} \) |
| 43 | \( 1 + 9.48T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 8.58T + 53T^{2} \) |
| 59 | \( 1 + 8.32T + 59T^{2} \) |
| 61 | \( 1 - 7.64T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 1.56T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 7.95T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.145260554684235120794160145658, −7.44240523983266285308329304773, −6.63801302107051940858403876576, −5.91329072295857248618417330415, −5.37203222264560764245729922760, −4.82134830222535610479615047770, −3.24605337574839866794103133928, −2.56335874175312005578140492899, −1.84159583966522427404212851071, −0.76302548766495202353506121798,
0.76302548766495202353506121798, 1.84159583966522427404212851071, 2.56335874175312005578140492899, 3.24605337574839866794103133928, 4.82134830222535610479615047770, 5.37203222264560764245729922760, 5.91329072295857248618417330415, 6.63801302107051940858403876576, 7.44240523983266285308329304773, 8.145260554684235120794160145658
