| L(s) = 1 | − 640.·3-s + 1.19e4·5-s + 1.74e4·7-s + 2.32e5·9-s + 5.42e5·11-s − 1.08e5·13-s − 7.65e6·15-s + 7.94e6·17-s − 7.86e6·19-s − 1.11e7·21-s + 3.44e7·23-s + 9.40e7·25-s − 3.55e7·27-s − 1.54e8·29-s + 5.14e7·31-s − 3.47e8·33-s + 2.08e8·35-s + 9.22e7·37-s + 6.92e7·39-s + 1.68e8·41-s − 2.39e8·43-s + 2.78e9·45-s − 6.90e7·47-s − 1.67e9·49-s − 5.08e9·51-s − 4.25e9·53-s + 6.48e9·55-s + ⋯ |
| L(s) = 1 | − 1.52·3-s + 1.71·5-s + 0.392·7-s + 1.31·9-s + 1.01·11-s − 0.0808·13-s − 2.60·15-s + 1.35·17-s − 0.729·19-s − 0.597·21-s + 1.11·23-s + 1.92·25-s − 0.476·27-s − 1.39·29-s + 0.323·31-s − 1.54·33-s + 0.671·35-s + 0.218·37-s + 0.122·39-s + 0.226·41-s − 0.248·43-s + 2.24·45-s − 0.0439·47-s − 0.845·49-s − 2.06·51-s − 1.39·53-s + 1.73·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.395564123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.395564123\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 640.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 1.19e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 1.74e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 5.42e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.08e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 7.94e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 7.86e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.44e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.54e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 5.14e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 9.22e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.68e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.39e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 6.90e7T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.25e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 5.88e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.82e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.15e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.55e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 7.70e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.66e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.50e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.50e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 4.20e10T + 7.15e21T^{2} \) |
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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
−18.57913133689240242696212205161, −17.24218712515558814080863642007, −16.89805231847574628656940167216, −14.41466031698629084608828705285, −12.69630395048473575487384499676, −11.08391566084478147030094759415, −9.642730463467278955859532244080, −6.43516722769812940935743573577, −5.26998016440216648596384796212, −1.35166993005988851330411790268,
1.35166993005988851330411790268, 5.26998016440216648596384796212, 6.43516722769812940935743573577, 9.642730463467278955859532244080, 11.08391566084478147030094759415, 12.69630395048473575487384499676, 14.41466031698629084608828705285, 16.89805231847574628656940167216, 17.24218712515558814080863642007, 18.57913133689240242696212205161
