| L(s) = 1 | − 0.406·3-s − 5-s − 2.91·7-s − 2.83·9-s − 6.51·11-s − 0.813·13-s + 0.406·15-s − 2.51·17-s − 0.406·19-s + 1.18·21-s − 5.02·23-s + 25-s + 2.37·27-s − 5.32·29-s + 8.75·31-s + 2.64·33-s + 2.91·35-s − 37-s + 0.330·39-s + 6.34·41-s + 7.32·43-s + 2.83·45-s + 5.42·47-s + 1.51·49-s + 1.02·51-s − 2.34·53-s + 6.51·55-s + ⋯ |
| L(s) = 1 | − 0.234·3-s − 0.447·5-s − 1.10·7-s − 0.944·9-s − 1.96·11-s − 0.225·13-s + 0.105·15-s − 0.608·17-s − 0.0933·19-s + 0.258·21-s − 1.04·23-s + 0.200·25-s + 0.456·27-s − 0.988·29-s + 1.57·31-s + 0.460·33-s + 0.493·35-s − 0.164·37-s + 0.0529·39-s + 0.990·41-s + 1.11·43-s + 0.422·45-s + 0.791·47-s + 0.215·49-s + 0.142·51-s − 0.322·53-s + 0.877·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3824897051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3824897051\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 0.406T + 3T^{2} \) |
| 7 | \( 1 + 2.91T + 7T^{2} \) |
| 11 | \( 1 + 6.51T + 11T^{2} \) |
| 13 | \( 1 + 0.813T + 13T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 19 | \( 1 + 0.406T + 19T^{2} \) |
| 23 | \( 1 + 5.02T + 23T^{2} \) |
| 29 | \( 1 + 5.32T + 29T^{2} \) |
| 31 | \( 1 - 8.75T + 31T^{2} \) |
| 41 | \( 1 - 6.34T + 41T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 - 1.42T + 59T^{2} \) |
| 61 | \( 1 + 1.32T + 61T^{2} \) |
| 67 | \( 1 - 5.42T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 1.75T + 79T^{2} \) |
| 83 | \( 1 - 7.05T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2.34T + 97T^{2} \) |
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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
−8.688240322991579673774987063749, −7.922142430062779106553121300300, −7.38662444324926802399908197144, −6.29729126556632529448683531508, −5.79331665913031198525451721955, −4.94484667962681739192639261692, −3.99414867640331656712284046560, −2.89886976444819861530483923934, −2.44703344706390752919430050567, −0.35000398300654225355479344215,
0.35000398300654225355479344215, 2.44703344706390752919430050567, 2.89886976444819861530483923934, 3.99414867640331656712284046560, 4.94484667962681739192639261692, 5.79331665913031198525451721955, 6.29729126556632529448683531508, 7.38662444324926802399908197144, 7.922142430062779106553121300300, 8.688240322991579673774987063749
