| L(s) = 1 | + 2.14·2-s + 2.87·3-s + 2.59·4-s + 6.16·6-s − 3.10·7-s + 1.26·8-s + 5.28·9-s − 1.10·11-s + 7.45·12-s − 1.77·13-s − 6.65·14-s − 2.47·16-s − 7.75·17-s + 11.3·18-s + 19-s − 8.93·21-s − 2.36·22-s + 6.65·23-s + 3.63·24-s − 3.79·26-s + 6.57·27-s − 8.04·28-s + 7.75·29-s + 6.57·31-s − 7.82·32-s − 3.18·33-s − 16.6·34-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.66·3-s + 1.29·4-s + 2.51·6-s − 1.17·7-s + 0.446·8-s + 1.76·9-s − 0.333·11-s + 2.15·12-s − 0.491·13-s − 1.77·14-s − 0.617·16-s − 1.88·17-s + 2.66·18-s + 0.229·19-s − 1.95·21-s − 0.504·22-s + 1.38·23-s + 0.742·24-s − 0.745·26-s + 1.26·27-s − 1.51·28-s + 1.44·29-s + 1.18·31-s − 1.38·32-s − 0.553·33-s − 2.84·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.288975375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.288975375\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 3 | \( 1 - 2.87T + 3T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 + 1.77T + 13T^{2} \) |
| 17 | \( 1 + 7.75T + 17T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 - 7.75T + 29T^{2} \) |
| 31 | \( 1 - 6.57T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 - 2.81T + 41T^{2} \) |
| 43 | \( 1 + 3.10T + 43T^{2} \) |
| 47 | \( 1 + 1.46T + 47T^{2} \) |
| 53 | \( 1 - 2.59T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 - 4.07T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 - 7.14T + 71T^{2} \) |
| 73 | \( 1 + 0.243T + 73T^{2} \) |
| 79 | \( 1 + 9.38T + 79T^{2} \) |
| 83 | \( 1 + 8.86T + 83T^{2} \) |
| 89 | \( 1 - 0.813T + 89T^{2} \) |
| 97 | \( 1 + 3.19T + 97T^{2} \) |
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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
−11.20239957506913439228659537596, −9.966452785722532456931604020407, −9.169833876468508841162276552094, −8.366950422454006530385427726972, −7.00563683171543417120531968409, −6.49472017848852510188881082873, −4.92466216797136882337836470725, −4.05262928251473097624945113995, −2.95177043823742787400304234771, −2.53338165698185548864165943671,
2.53338165698185548864165943671, 2.95177043823742787400304234771, 4.05262928251473097624945113995, 4.92466216797136882337836470725, 6.49472017848852510188881082873, 7.00563683171543417120531968409, 8.366950422454006530385427726972, 9.169833876468508841162276552094, 9.966452785722532456931604020407, 11.20239957506913439228659537596
