| L(s) = 1 | − 2.04·2-s − 3-s + 2.18·4-s + 2.04·6-s + 1.36·7-s − 0.377·8-s + 9-s − 6.09·11-s − 2.18·12-s − 1.01·13-s − 2.78·14-s − 3.59·16-s + 3.16·17-s − 2.04·18-s − 2.69·19-s − 1.36·21-s + 12.4·22-s + 8.33·23-s + 0.377·24-s + 2.07·26-s − 27-s + 2.97·28-s + 8.92·29-s − 6.61·31-s + 8.11·32-s + 6.09·33-s − 6.46·34-s + ⋯ |
| L(s) = 1 | − 1.44·2-s − 0.577·3-s + 1.09·4-s + 0.835·6-s + 0.515·7-s − 0.133·8-s + 0.333·9-s − 1.83·11-s − 0.630·12-s − 0.281·13-s − 0.745·14-s − 0.899·16-s + 0.766·17-s − 0.482·18-s − 0.617·19-s − 0.297·21-s + 2.66·22-s + 1.73·23-s + 0.0770·24-s + 0.407·26-s − 0.192·27-s + 0.562·28-s + 1.65·29-s − 1.18·31-s + 1.43·32-s + 1.06·33-s − 1.10·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 2.04T + 2T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 + 6.09T + 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 - 8.33T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 + 6.61T + 31T^{2} \) |
| 37 | \( 1 + 0.367T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 + 8.55T + 43T^{2} \) |
| 53 | \( 1 - 0.984T + 53T^{2} \) |
| 59 | \( 1 + 1.26T + 59T^{2} \) |
| 61 | \( 1 - 3.49T + 61T^{2} \) |
| 67 | \( 1 + 9.10T + 67T^{2} \) |
| 71 | \( 1 + 9.72T + 71T^{2} \) |
| 73 | \( 1 - 6.26T + 73T^{2} \) |
| 79 | \( 1 - 8.41T + 79T^{2} \) |
| 83 | \( 1 + 1.75T + 83T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−8.325656491680844643887524069358, −7.51621381136935570101627689534, −7.13592140719268246230773626932, −6.09871895898684325841831099259, −5.03226712059720438059683685917, −4.77337325927818628846332750788, −3.14511984262275692621296484328, −2.18556626009350893630432512441, −1.09762008334160691404717546246, 0,
1.09762008334160691404717546246, 2.18556626009350893630432512441, 3.14511984262275692621296484328, 4.77337325927818628846332750788, 5.03226712059720438059683685917, 6.09871895898684325841831099259, 7.13592140719268246230773626932, 7.51621381136935570101627689534, 8.325656491680844643887524069358
