| L(s) = 1 | + 3.43·3-s − 0.723·5-s − 3.63·7-s + 8.77·9-s + 2.52·11-s − 5.63·13-s − 2.48·15-s − 6.45·17-s − 6.03·19-s − 12.4·21-s − 6.51·23-s − 4.47·25-s + 19.7·27-s + 1.42·29-s − 1.46·31-s + 8.66·33-s + 2.63·35-s + 9.25·37-s − 19.3·39-s − 3.54·41-s − 2.04·43-s − 6.34·45-s + 1.56·47-s + 6.24·49-s − 22.1·51-s + 0.205·53-s − 1.82·55-s + ⋯ |
| L(s) = 1 | + 1.98·3-s − 0.323·5-s − 1.37·7-s + 2.92·9-s + 0.761·11-s − 1.56·13-s − 0.641·15-s − 1.56·17-s − 1.38·19-s − 2.72·21-s − 1.35·23-s − 0.895·25-s + 3.81·27-s + 0.264·29-s − 0.263·31-s + 1.50·33-s + 0.445·35-s + 1.52·37-s − 3.09·39-s − 0.553·41-s − 0.312·43-s − 0.946·45-s + 0.229·47-s + 0.891·49-s − 3.09·51-s + 0.0282·53-s − 0.246·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 3.43T + 3T^{2} \) |
| 5 | \( 1 + 0.723T + 5T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 - 2.52T + 11T^{2} \) |
| 13 | \( 1 + 5.63T + 13T^{2} \) |
| 17 | \( 1 + 6.45T + 17T^{2} \) |
| 19 | \( 1 + 6.03T + 19T^{2} \) |
| 23 | \( 1 + 6.51T + 23T^{2} \) |
| 29 | \( 1 - 1.42T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 9.25T + 37T^{2} \) |
| 41 | \( 1 + 3.54T + 41T^{2} \) |
| 43 | \( 1 + 2.04T + 43T^{2} \) |
| 47 | \( 1 - 1.56T + 47T^{2} \) |
| 53 | \( 1 - 0.205T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 3.53T + 61T^{2} \) |
| 67 | \( 1 + 3.41T + 67T^{2} \) |
| 71 | \( 1 + 8.33T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 8.98T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−8.086788888686415259828165145914, −7.52823458926273640505223390099, −6.71296254773044453814304942848, −6.29206460784313714049513146043, −4.47737543582869826325277029172, −4.17459859779332122262432520453, −3.34168018398063966447641138784, −2.46268680656422246334899328340, −1.97977689628397592383325537243, 0,
1.97977689628397592383325537243, 2.46268680656422246334899328340, 3.34168018398063966447641138784, 4.17459859779332122262432520453, 4.47737543582869826325277029172, 6.29206460784313714049513146043, 6.71296254773044453814304942848, 7.52823458926273640505223390099, 8.086788888686415259828165145914
