| L(s) = 1 | + 37.1·2-s + 864.·4-s − 1.84e3·5-s − 2.40e3·7-s + 1.30e4·8-s − 6.82e4·10-s + 4.02e4·11-s − 7.54e4·13-s − 8.90e4·14-s + 4.30e4·16-s − 5.50e5·17-s − 6.61e5·19-s − 1.59e6·20-s + 1.49e6·22-s − 1.03e6·23-s + 1.43e6·25-s − 2.80e6·26-s − 2.07e6·28-s + 2.37e6·29-s + 2.81e6·31-s − 5.10e6·32-s − 2.04e7·34-s + 4.41e6·35-s − 1.64e7·37-s − 2.45e7·38-s − 2.40e7·40-s + 2.33e7·41-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + 1.68·4-s − 1.31·5-s − 0.377·7-s + 1.13·8-s − 2.15·10-s + 0.829·11-s − 0.732·13-s − 0.619·14-s + 0.164·16-s − 1.59·17-s − 1.16·19-s − 2.22·20-s + 1.36·22-s − 0.767·23-s + 0.734·25-s − 1.20·26-s − 0.638·28-s + 0.624·29-s + 0.547·31-s − 0.861·32-s − 2.62·34-s + 0.497·35-s − 1.44·37-s − 1.90·38-s − 1.48·40-s + 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 - 37.1T + 512T^{2} \) |
| 5 | \( 1 + 1.84e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 4.02e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.54e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.50e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.61e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.03e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.37e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.81e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.64e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.33e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.96e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.95e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.06e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.34e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 7.23e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.02e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.00e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.40e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.38e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.03e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.05e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.10e9T + 7.60e17T^{2} \) |
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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
−12.41916193308978136759295728522, −11.86842720947699392957942432277, −10.77418653974907194696332660518, −8.813868945641497665127442251474, −7.21352209442377475218051843719, −6.23885949501540103212837564158, −4.49802060359616061135204673871, −3.92643370139946868812081051585, −2.45177788380206353062604377161, 0,
2.45177788380206353062604377161, 3.92643370139946868812081051585, 4.49802060359616061135204673871, 6.23885949501540103212837564158, 7.21352209442377475218051843719, 8.813868945641497665127442251474, 10.77418653974907194696332660518, 11.86842720947699392957942432277, 12.41916193308978136759295728522
