| L(s) = 1 | − 918.·5-s + 6.35e3·7-s + 3.08e4·11-s − 6.16e4·13-s − 2.05e5·17-s + 1.55e5·19-s + 4.29e5·23-s − 1.10e6·25-s − 5.02e6·29-s + 1.80e6·31-s − 5.83e6·35-s + 7.46e6·37-s − 1.52e7·41-s + 3.56e7·43-s + 1.70e7·47-s + 6.52e4·49-s + 5.94e7·53-s − 2.83e7·55-s − 6.14e5·59-s − 6.24e7·61-s + 5.66e7·65-s + 1.38e8·67-s − 1.37e8·71-s + 4.54e7·73-s + 1.95e8·77-s − 3.81e8·79-s + 2.48e7·83-s + ⋯ |
| L(s) = 1 | − 0.657·5-s + 1.00·7-s + 0.634·11-s − 0.598·13-s − 0.596·17-s + 0.274·19-s + 0.319·23-s − 0.568·25-s − 1.31·29-s + 0.350·31-s − 0.657·35-s + 0.654·37-s − 0.841·41-s + 1.58·43-s + 0.508·47-s + 0.00161·49-s + 1.03·53-s − 0.417·55-s − 0.00660·59-s − 0.577·61-s + 0.393·65-s + 0.839·67-s − 0.642·71-s + 0.187·73-s + 0.635·77-s − 1.10·79-s + 0.0575·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 918.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.35e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.08e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.16e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.05e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.55e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 4.29e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.02e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.80e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.46e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.52e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.56e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.70e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.94e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.14e5T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.24e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.38e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.37e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.54e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.81e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.48e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.76e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.55e7T + 7.60e17T^{2} \) |
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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
−9.688163171742520884273808718674, −8.748839785602149451673795367833, −7.79246752711959689925820647497, −7.07297444231174710839851798354, −5.73121175412596553525994134195, −4.62396599350025127171247792259, −3.82557784967102944329888861087, −2.38978843096871286307679926026, −1.25559596492856674127875013140, 0,
1.25559596492856674127875013140, 2.38978843096871286307679926026, 3.82557784967102944329888861087, 4.62396599350025127171247792259, 5.73121175412596553525994134195, 7.07297444231174710839851798354, 7.79246752711959689925820647497, 8.748839785602149451673795367833, 9.688163171742520884273808718674
