| L(s) = 1 | + 1.55·2-s − 4.96·3-s − 5.58·4-s − 7.71·6-s + 7·7-s − 21.1·8-s − 2.38·9-s + 29.8·11-s + 27.6·12-s + 90.7·13-s + 10.8·14-s + 11.7·16-s + 29.5·17-s − 3.70·18-s − 62.3·19-s − 34.7·21-s + 46.3·22-s + 90.6·23-s + 104.·24-s + 141.·26-s + 145.·27-s − 39.0·28-s + 193.·29-s − 152.·31-s + 187.·32-s − 147.·33-s + 45.9·34-s + ⋯ |
| L(s) = 1 | + 0.549·2-s − 0.954·3-s − 0.697·4-s − 0.525·6-s + 0.377·7-s − 0.933·8-s − 0.0882·9-s + 0.817·11-s + 0.666·12-s + 1.93·13-s + 0.207·14-s + 0.184·16-s + 0.421·17-s − 0.0485·18-s − 0.752·19-s − 0.360·21-s + 0.449·22-s + 0.821·23-s + 0.891·24-s + 1.06·26-s + 1.03·27-s − 0.263·28-s + 1.23·29-s − 0.881·31-s + 1.03·32-s − 0.780·33-s + 0.232·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.392043206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.392043206\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 1.55T + 8T^{2} \) |
| 3 | \( 1 + 4.96T + 27T^{2} \) |
| 11 | \( 1 - 29.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 90.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 62.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 90.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 193.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 102.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 387.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 152.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 81.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 235.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 510.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 347.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 317.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 709.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 503.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 482.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 481.T + 9.12e5T^{2} \) |
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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
−12.24564744287305817197683230721, −11.37700675714668244784993218792, −10.59071143359218619923739112422, −9.081497581070433486425702362394, −8.364339837620896382315771286701, −6.49656153616018966220551853942, −5.77818845190338978788656727815, −4.65521735189639904497631826090, −3.50941245521523224717411538051, −0.953443287762168989611222339201,
0.953443287762168989611222339201, 3.50941245521523224717411538051, 4.65521735189639904497631826090, 5.77818845190338978788656727815, 6.49656153616018966220551853942, 8.364339837620896382315771286701, 9.081497581070433486425702362394, 10.59071143359218619923739112422, 11.37700675714668244784993218792, 12.24564744287305817197683230721
