| L(s) = 1 | + 7.08·2-s − 10.0·3-s + 18.1·4-s − 79.8·5-s − 71.4·6-s − 97.9·8-s − 141.·9-s − 565.·10-s + 351.·11-s − 183.·12-s − 291.·13-s + 804.·15-s − 1.27e3·16-s − 370.·17-s − 1.00e3·18-s + 1.50e3·19-s − 1.45e3·20-s + 2.49e3·22-s − 425.·23-s + 987.·24-s + 3.24e3·25-s − 2.06e3·26-s + 3.87e3·27-s − 7.78e3·29-s + 5.70e3·30-s − 2.57e3·31-s − 5.89e3·32-s + ⋯ |
| L(s) = 1 | + 1.25·2-s − 0.646·3-s + 0.567·4-s − 1.42·5-s − 0.809·6-s − 0.541·8-s − 0.581·9-s − 1.78·10-s + 0.876·11-s − 0.367·12-s − 0.478·13-s + 0.923·15-s − 1.24·16-s − 0.310·17-s − 0.728·18-s + 0.956·19-s − 0.810·20-s + 1.09·22-s − 0.167·23-s + 0.350·24-s + 1.03·25-s − 0.599·26-s + 1.02·27-s − 1.71·29-s + 1.15·30-s − 0.481·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 7.08T + 32T^{2} \) |
| 3 | \( 1 + 10.0T + 243T^{2} \) |
| 5 | \( 1 + 79.8T + 3.12e3T^{2} \) |
| 11 | \( 1 - 351.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 291.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 370.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 425.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 739.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.02e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.83e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.53e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.53e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.96e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.65e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−14.14430956103366514539280512831, −12.67105801810661136471000522684, −11.76842490935789868639683464795, −11.26116481820268018581504288932, −9.071531298190257858435339852059, −7.39900359603591289223417542366, −5.90616051394942511995163511068, −4.56893824872503263530190255472, −3.38413568468120653904066548416, 0,
3.38413568468120653904066548416, 4.56893824872503263530190255472, 5.90616051394942511995163511068, 7.39900359603591289223417542366, 9.071531298190257858435339852059, 11.26116481820268018581504288932, 11.76842490935789868639683464795, 12.67105801810661136471000522684, 14.14430956103366514539280512831
