| L(s) = 1 | − 2-s − 2.11·3-s + 4-s − 3.30·5-s + 2.11·6-s + 7-s − 8-s + 1.47·9-s + 3.30·10-s − 0.751·11-s − 2.11·12-s + 0.668·13-s − 14-s + 6.98·15-s + 16-s − 6.66·17-s − 1.47·18-s + 3.47·19-s − 3.30·20-s − 2.11·21-s + 0.751·22-s + 4.23·23-s + 2.11·24-s + 5.90·25-s − 0.668·26-s + 3.21·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.22·3-s + 0.5·4-s − 1.47·5-s + 0.864·6-s + 0.377·7-s − 0.353·8-s + 0.493·9-s + 1.04·10-s − 0.226·11-s − 0.610·12-s + 0.185·13-s − 0.267·14-s + 1.80·15-s + 0.250·16-s − 1.61·17-s − 0.348·18-s + 0.796·19-s − 0.738·20-s − 0.461·21-s + 0.160·22-s + 0.882·23-s + 0.432·24-s + 1.18·25-s − 0.131·26-s + 0.619·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 2.11T + 3T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 11 | \( 1 + 0.751T + 11T^{2} \) |
| 13 | \( 1 - 0.668T + 13T^{2} \) |
| 17 | \( 1 + 6.66T + 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 23 | \( 1 - 4.23T + 23T^{2} \) |
| 29 | \( 1 + 8.57T + 29T^{2} \) |
| 31 | \( 1 + 2.10T + 31T^{2} \) |
| 37 | \( 1 + 4.49T + 37T^{2} \) |
| 41 | \( 1 - 0.614T + 41T^{2} \) |
| 43 | \( 1 + 0.977T + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 + 6.80T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 5.98T + 67T^{2} \) |
| 71 | \( 1 + 1.53T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 7.79T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 2.73T + 89T^{2} \) |
| 97 | \( 1 - 9.43T + 97T^{2} \) |
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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
−7.67929312853440048679166402519, −7.01952343580150989949263850001, −6.58488872628012098825540493331, −5.48385168249327686976995287863, −4.99428472248197810891096159419, −4.09249352596100771025645244103, −3.32262187063816658032269384180, −2.09228738045522362467902126895, −0.812928427625931221337758205872, 0,
0.812928427625931221337758205872, 2.09228738045522362467902126895, 3.32262187063816658032269384180, 4.09249352596100771025645244103, 4.99428472248197810891096159419, 5.48385168249327686976995287863, 6.58488872628012098825540493331, 7.01952343580150989949263850001, 7.67929312853440048679166402519
