| L(s) = 1 | − 1.07·2-s + 1.06·3-s − 0.833·4-s − 1.15·6-s − 2.95·7-s + 3.06·8-s − 1.85·9-s + 5.89·11-s − 0.891·12-s − 3.64·13-s + 3.18·14-s − 1.63·16-s + 4.68·17-s + 2.00·18-s + 5.73·19-s − 3.15·21-s − 6.36·22-s + 23-s + 3.27·24-s + 3.94·26-s − 5.19·27-s + 2.46·28-s + 10.3·29-s + 7.29·31-s − 4.35·32-s + 6.30·33-s − 5.06·34-s + ⋯ |
| L(s) = 1 | − 0.763·2-s + 0.617·3-s − 0.416·4-s − 0.471·6-s − 1.11·7-s + 1.08·8-s − 0.618·9-s + 1.77·11-s − 0.257·12-s − 1.01·13-s + 0.852·14-s − 0.409·16-s + 1.13·17-s + 0.472·18-s + 1.31·19-s − 0.689·21-s − 1.35·22-s + 0.208·23-s + 0.667·24-s + 0.772·26-s − 0.999·27-s + 0.465·28-s + 1.92·29-s + 1.31·31-s − 0.769·32-s + 1.09·33-s − 0.867·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9728289617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9728289617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.07T + 2T^{2} \) |
| 3 | \( 1 - 1.06T + 3T^{2} \) |
| 7 | \( 1 + 2.95T + 7T^{2} \) |
| 11 | \( 1 - 5.89T + 11T^{2} \) |
| 13 | \( 1 + 3.64T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 7.29T + 31T^{2} \) |
| 37 | \( 1 + 0.612T + 37T^{2} \) |
| 41 | \( 1 + 9.10T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 - 3.17T + 53T^{2} \) |
| 59 | \( 1 - 1.38T + 59T^{2} \) |
| 61 | \( 1 - 6.02T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 6.22T + 71T^{2} \) |
| 73 | \( 1 + 4.48T + 73T^{2} \) |
| 79 | \( 1 + 7.01T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 - 2.15T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−10.11642207749180733412884930222, −9.778147306155293680841031657220, −9.048357206341576456153807063247, −8.337697138837693863309626921763, −7.32746320621435569463463494153, −6.43019975595639224384247930632, −5.12505070508469272458451546294, −3.81219646137762222784405398822, −2.88667435643997365809850131275, −0.996550874545041937901103527487,
0.996550874545041937901103527487, 2.88667435643997365809850131275, 3.81219646137762222784405398822, 5.12505070508469272458451546294, 6.43019975595639224384247930632, 7.32746320621435569463463494153, 8.337697138837693863309626921763, 9.048357206341576456153807063247, 9.778147306155293680841031657220, 10.11642207749180733412884930222
