L(s) = 1 | + 2-s + 0.529·3-s + 4-s + 0.529·6-s − 7-s + 8-s − 2.71·9-s + 11-s + 0.529·12-s + 2.71·13-s − 14-s + 16-s + 2.52·17-s − 2.71·18-s − 1.77·19-s − 0.529·21-s + 22-s − 6.27·23-s + 0.529·24-s + 2.71·26-s − 3.02·27-s − 28-s + 7.33·29-s + 5.36·31-s + 32-s + 0.529·33-s + 2.52·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.305·3-s + 0.5·4-s + 0.216·6-s − 0.377·7-s + 0.353·8-s − 0.906·9-s + 0.301·11-s + 0.152·12-s + 0.754·13-s − 0.267·14-s + 0.250·16-s + 0.613·17-s − 0.641·18-s − 0.408·19-s − 0.115·21-s + 0.213·22-s − 1.30·23-s + 0.108·24-s + 0.533·26-s − 0.582·27-s − 0.188·28-s + 1.36·29-s + 0.963·31-s + 0.176·32-s + 0.0921·33-s + 0.433·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.226607649\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.226607649\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 0.529T + 3T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 + 1.77T + 19T^{2} \) |
| 23 | \( 1 + 6.27T + 23T^{2} \) |
| 29 | \( 1 - 7.33T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 8.99T + 41T^{2} \) |
| 43 | \( 1 - 1.94T + 43T^{2} \) |
| 47 | \( 1 + 1.96T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 7.02T + 59T^{2} \) |
| 61 | \( 1 - 0.443T + 61T^{2} \) |
| 67 | \( 1 + 0.941T + 67T^{2} \) |
| 71 | \( 1 - 1.16T + 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 + 3.22T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 9.77T + 89T^{2} \) |
| 97 | \( 1 - 1.16T + 97T^{2} \) |
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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
−8.261214927446294291817606481258, −7.935540767523949312469357375097, −6.80542571288583062425101106816, −6.03911210518406873040814796746, −5.76304636250686467061952429203, −4.51284439204362721226014059822, −3.90242813840368687485290448795, −2.98971071466105910403455534270, −2.34422788019405519528201713629, −0.933053065227256125194141120400,
0.933053065227256125194141120400, 2.34422788019405519528201713629, 2.98971071466105910403455534270, 3.90242813840368687485290448795, 4.51284439204362721226014059822, 5.76304636250686467061952429203, 6.03911210518406873040814796746, 6.80542571288583062425101106816, 7.935540767523949312469357375097, 8.261214927446294291817606481258