| L(s) = 1 | − 16·2-s + 122.·3-s + 256·4-s + 625·5-s − 1.95e3·6-s + 2.40e3·7-s − 4.09e3·8-s − 4.78e3·9-s − 1.00e4·10-s − 918.·11-s + 3.12e4·12-s − 1.74e5·13-s − 3.84e4·14-s + 7.62e4·15-s + 6.55e4·16-s − 1.57e5·17-s + 7.66e4·18-s − 8.50e5·19-s + 1.60e5·20-s + 2.93e5·21-s + 1.46e4·22-s + 1.18e6·23-s − 4.99e5·24-s + 3.90e5·25-s + 2.79e6·26-s − 2.98e6·27-s + 6.14e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.869·3-s + 0.5·4-s + 0.447·5-s − 0.615·6-s + 0.377·7-s − 0.353·8-s − 0.243·9-s − 0.316·10-s − 0.0189·11-s + 0.434·12-s − 1.69·13-s − 0.267·14-s + 0.389·15-s + 0.250·16-s − 0.458·17-s + 0.172·18-s − 1.49·19-s + 0.223·20-s + 0.328·21-s + 0.0133·22-s + 0.880·23-s − 0.307·24-s + 0.200·25-s + 1.19·26-s − 1.08·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 16T \) |
| 5 | \( 1 - 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 3 | \( 1 - 122.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 918.T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.74e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.57e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.50e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.18e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.21e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.33e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.30e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.97e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.02e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.68e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.35e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.29e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.47e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 5.26e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.22e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 8.11e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.33e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.52e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.70e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.59e8T + 7.60e17T^{2} \) |
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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
−12.22020221034861508969206577434, −10.89607070705651888737141704476, −9.750808864962110761526617728054, −8.807772719875007416171798406015, −7.86151769363163172762491120217, −6.56551907139892039272241082782, −4.83143322997916784309353268561, −2.84958492120401388139472899737, −1.91027291184698527997534179971, 0,
1.91027291184698527997534179971, 2.84958492120401388139472899737, 4.83143322997916784309353268561, 6.56551907139892039272241082782, 7.86151769363163172762491120217, 8.807772719875007416171798406015, 9.750808864962110761526617728054, 10.89607070705651888737141704476, 12.22020221034861508969206577434
