| L(s) = 1 | − 1.54·2-s + 27·3-s − 125.·4-s − 293.·5-s − 41.7·6-s + 391.·8-s + 729·9-s + 453.·10-s + 3.15e3·11-s − 3.39e3·12-s + 9.24e3·13-s − 7.93e3·15-s + 1.54e4·16-s + 1.98e4·17-s − 1.12e3·18-s − 1.35e4·19-s + 3.68e4·20-s − 4.86e3·22-s − 3.51e4·23-s + 1.05e4·24-s + 8.14e3·25-s − 1.42e4·26-s + 1.96e4·27-s − 1.41e5·29-s + 1.22e4·30-s − 2.30e5·31-s − 7.40e4·32-s + ⋯ |
| L(s) = 1 | − 0.136·2-s + 0.577·3-s − 0.981·4-s − 1.05·5-s − 0.0788·6-s + 0.270·8-s + 0.333·9-s + 0.143·10-s + 0.713·11-s − 0.566·12-s + 1.16·13-s − 0.606·15-s + 0.944·16-s + 0.981·17-s − 0.0455·18-s − 0.452·19-s + 1.03·20-s − 0.0974·22-s − 0.602·23-s + 0.156·24-s + 0.104·25-s − 0.159·26-s + 0.192·27-s − 1.07·29-s + 0.0828·30-s − 1.38·31-s − 0.399·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.54T + 128T^{2} \) |
| 5 | \( 1 + 293.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 3.15e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.24e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.98e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.35e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.51e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.30e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.43e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.41e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.08e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.73e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.60e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.21e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.55e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.41e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.39e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.52e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.31e6T + 8.07e13T^{2} \) |
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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
−11.24197131221626143012245386595, −10.01539088840288081492088022469, −8.916307100387221745933443721747, −8.268550405012724099237418850648, −7.27101607861222669432402062534, −5.61648159315584758114830792001, −4.00395058243036648731223185895, −3.61564947541806551875378688230, −1.41093584789981013389528652268, 0,
1.41093584789981013389528652268, 3.61564947541806551875378688230, 4.00395058243036648731223185895, 5.61648159315584758114830792001, 7.27101607861222669432402062534, 8.268550405012724099237418850648, 8.916307100387221745933443721747, 10.01539088840288081492088022469, 11.24197131221626143012245386595
