| L(s) = 1 | + 3-s + 5-s + 2.50·7-s + 9-s + 2.50·11-s − 13-s + 15-s − 5.27·17-s − 3.23·19-s + 2.50·21-s − 3.27·23-s + 25-s + 27-s − 7.00·29-s + 9.77·31-s + 2.50·33-s + 2.50·35-s + 5.27·37-s − 39-s + 11.7·41-s + 9.00·43-s + 45-s − 5.77·47-s − 0.733·49-s − 5.27·51-s + 2.72·53-s + 2.50·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.946·7-s + 0.333·9-s + 0.754·11-s − 0.277·13-s + 0.258·15-s − 1.27·17-s − 0.741·19-s + 0.546·21-s − 0.682·23-s + 0.200·25-s + 0.192·27-s − 1.30·29-s + 1.75·31-s + 0.435·33-s + 0.423·35-s + 0.866·37-s − 0.160·39-s + 1.83·41-s + 1.37·43-s + 0.149·45-s − 0.842·47-s − 0.104·49-s − 0.738·51-s + 0.374·53-s + 0.337·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.190582076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.190582076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 - 2.50T + 11T^{2} \) |
| 17 | \( 1 + 5.27T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 + 3.27T + 23T^{2} \) |
| 29 | \( 1 + 7.00T + 29T^{2} \) |
| 31 | \( 1 - 9.77T + 31T^{2} \) |
| 37 | \( 1 - 5.27T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 9.00T + 43T^{2} \) |
| 47 | \( 1 + 5.77T + 47T^{2} \) |
| 53 | \( 1 - 2.72T + 53T^{2} \) |
| 59 | \( 1 - 5.77T + 59T^{2} \) |
| 61 | \( 1 - 9.27T + 61T^{2} \) |
| 67 | \( 1 + 1.77T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 0.726T + 79T^{2} \) |
| 83 | \( 1 - 1.77T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−8.000665346969985036829829536062, −7.54420722221040656702253361032, −6.51478501605931146222863840122, −6.12682841889645545559456534178, −5.01849667978628796528617820788, −4.36247567850828928003541356113, −3.79466137559504576528199137847, −2.40401957070359678504075015187, −2.11286667597339434887751903435, −0.929929148884187071246903180701,
0.929929148884187071246903180701, 2.11286667597339434887751903435, 2.40401957070359678504075015187, 3.79466137559504576528199137847, 4.36247567850828928003541356113, 5.01849667978628796528617820788, 6.12682841889645545559456534178, 6.51478501605931146222863840122, 7.54420722221040656702253361032, 8.000665346969985036829829536062
