| L(s) = 1 | + 2.23·2-s − 0.740·3-s + 3.01·4-s − 1.65·6-s + 3.67·7-s + 2.27·8-s − 2.45·9-s + 5.02·11-s − 2.23·12-s − 2.67·13-s + 8.23·14-s − 0.936·16-s + 3.30·17-s − 5.48·18-s + 2.88·19-s − 2.72·21-s + 11.2·22-s + 6.01·23-s − 1.68·24-s − 5.98·26-s + 4.03·27-s + 11.0·28-s − 5.13·29-s − 31-s − 6.64·32-s − 3.72·33-s + 7.40·34-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.427·3-s + 1.50·4-s − 0.677·6-s + 1.38·7-s + 0.804·8-s − 0.817·9-s + 1.51·11-s − 0.644·12-s − 0.741·13-s + 2.19·14-s − 0.234·16-s + 0.801·17-s − 1.29·18-s + 0.660·19-s − 0.594·21-s + 2.39·22-s + 1.25·23-s − 0.344·24-s − 1.17·26-s + 0.777·27-s + 2.09·28-s − 0.952·29-s − 0.179·31-s − 1.17·32-s − 0.648·33-s + 1.26·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.563360447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.563360447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 3 | \( 1 + 0.740T + 3T^{2} \) |
| 7 | \( 1 - 3.67T + 7T^{2} \) |
| 11 | \( 1 - 5.02T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 - 3.30T + 17T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 23 | \( 1 - 6.01T + 23T^{2} \) |
| 29 | \( 1 + 5.13T + 29T^{2} \) |
| 37 | \( 1 + 7.90T + 37T^{2} \) |
| 41 | \( 1 + 9.55T + 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 + 1.06T + 47T^{2} \) |
| 53 | \( 1 + 0.521T + 53T^{2} \) |
| 59 | \( 1 + 8.09T + 59T^{2} \) |
| 61 | \( 1 - 0.598T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 8.80T + 71T^{2} \) |
| 73 | \( 1 - 9.87T + 73T^{2} \) |
| 79 | \( 1 + 2.40T + 79T^{2} \) |
| 83 | \( 1 + 2.91T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−10.89968539088231830645258779760, −9.457714755356255354056827070814, −8.546180871874032442360219079801, −7.36161147581416378243227382199, −6.57324457494254064941566794634, −5.37522379329310421644097021373, −5.15739748186812638570123385470, −4.03695972713154222611959678039, −3.03668109775885583481354355607, −1.58125996870156847299024741641,
1.58125996870156847299024741641, 3.03668109775885583481354355607, 4.03695972713154222611959678039, 5.15739748186812638570123385470, 5.37522379329310421644097021373, 6.57324457494254064941566794634, 7.36161147581416378243227382199, 8.546180871874032442360219079801, 9.457714755356255354056827070814, 10.89968539088231830645258779760
