| L(s) = 1 | − 2-s − 0.531·3-s + 4-s + 1.86·5-s + 0.531·6-s − 4.06·7-s − 8-s − 2.71·9-s − 1.86·10-s + 2.15·11-s − 0.531·12-s + 0.240·13-s + 4.06·14-s − 0.990·15-s + 16-s + 5.05·17-s + 2.71·18-s − 2.35·19-s + 1.86·20-s + 2.15·21-s − 2.15·22-s + 6.69·23-s + 0.531·24-s − 1.52·25-s − 0.240·26-s + 3.03·27-s − 4.06·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.306·3-s + 0.5·4-s + 0.833·5-s + 0.216·6-s − 1.53·7-s − 0.353·8-s − 0.905·9-s − 0.589·10-s + 0.649·11-s − 0.153·12-s + 0.0667·13-s + 1.08·14-s − 0.255·15-s + 0.250·16-s + 1.22·17-s + 0.640·18-s − 0.540·19-s + 0.416·20-s + 0.471·21-s − 0.459·22-s + 1.39·23-s + 0.108·24-s − 0.305·25-s − 0.0472·26-s + 0.584·27-s − 0.768·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 0.531T + 3T^{2} \) |
| 5 | \( 1 - 1.86T + 5T^{2} \) |
| 7 | \( 1 + 4.06T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 - 0.240T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 23 | \( 1 - 6.69T + 23T^{2} \) |
| 29 | \( 1 + 7.35T + 29T^{2} \) |
| 37 | \( 1 - 1.51T + 37T^{2} \) |
| 41 | \( 1 - 0.437T + 41T^{2} \) |
| 43 | \( 1 + 2.73T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 9.67T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 - 3.34T + 67T^{2} \) |
| 71 | \( 1 - 1.95T + 71T^{2} \) |
| 73 | \( 1 - 1.25T + 73T^{2} \) |
| 79 | \( 1 - 6.32T + 79T^{2} \) |
| 83 | \( 1 - 9.96T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
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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
−7.79491505595938308343630609005, −6.67960189265765563755870076458, −6.55355260858653120808370769573, −5.71028357611844641931365015725, −5.22397765385211198660529156969, −3.71015070110569353240165962148, −3.15436585213835409641471482135, −2.27928932144524930976572502124, −1.13720242990989978609238318666, 0,
1.13720242990989978609238318666, 2.27928932144524930976572502124, 3.15436585213835409641471482135, 3.71015070110569353240165962148, 5.22397765385211198660529156969, 5.71028357611844641931365015725, 6.55355260858653120808370769573, 6.67960189265765563755870076458, 7.79491505595938308343630609005
