| L(s) = 1 | + 2-s + 2.59·3-s + 4-s + 5-s + 2.59·6-s + 1.59·7-s + 8-s + 3.73·9-s + 10-s − 1.73·11-s + 2.59·12-s + 4.34·13-s + 1.59·14-s + 2.59·15-s + 16-s + 1.14·17-s + 3.73·18-s − 3.54·19-s + 20-s + 4.15·21-s − 1.73·22-s + 2.59·24-s + 25-s + 4.34·26-s + 1.91·27-s + 1.59·28-s − 7.51·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.49·3-s + 0.5·4-s + 0.447·5-s + 1.05·6-s + 0.604·7-s + 0.353·8-s + 1.24·9-s + 0.316·10-s − 0.522·11-s + 0.749·12-s + 1.20·13-s + 0.427·14-s + 0.670·15-s + 0.250·16-s + 0.277·17-s + 0.881·18-s − 0.813·19-s + 0.223·20-s + 0.905·21-s − 0.369·22-s + 0.529·24-s + 0.200·25-s + 0.852·26-s + 0.369·27-s + 0.302·28-s − 1.39·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.664049876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.664049876\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.59T + 3T^{2} \) |
| 7 | \( 1 - 1.59T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 29 | \( 1 + 7.51T + 29T^{2} \) |
| 31 | \( 1 - 6.58T + 31T^{2} \) |
| 37 | \( 1 + 5.43T + 37T^{2} \) |
| 41 | \( 1 - 2.66T + 41T^{2} \) |
| 43 | \( 1 - 6.62T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 1.94T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 4.94T + 61T^{2} \) |
| 67 | \( 1 - 4.62T + 67T^{2} \) |
| 71 | \( 1 - 7.49T + 71T^{2} \) |
| 73 | \( 1 - 3.24T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 3.76T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 5.13T + 97T^{2} \) |
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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
−8.314540795522136848494732018431, −7.57025892800081071374202192382, −6.83867039038478934537536778422, −5.90100319946608326095786340794, −5.29985452579944586301271986735, −4.19851694295643419549762789884, −3.76716009926277039579539771688, −2.78993013560446035973773772653, −2.19422607909032471388960102225, −1.32515228584769609930688632826,
1.32515228584769609930688632826, 2.19422607909032471388960102225, 2.78993013560446035973773772653, 3.76716009926277039579539771688, 4.19851694295643419549762789884, 5.29985452579944586301271986735, 5.90100319946608326095786340794, 6.83867039038478934537536778422, 7.57025892800081071374202192382, 8.314540795522136848494732018431
