| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 2.44·7-s − 8-s + 9-s − 4.89·11-s + 12-s + 6·13-s + 2.44·14-s + 16-s + 17-s − 18-s + 6.89·19-s − 2.44·21-s + 4.89·22-s + 6.44·23-s − 24-s − 6·26-s + 27-s − 2.44·28-s − 9.34·29-s − 6.44·31-s − 32-s − 4.89·33-s − 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.925·7-s − 0.353·8-s + 0.333·9-s − 1.47·11-s + 0.288·12-s + 1.66·13-s + 0.654·14-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.58·19-s − 0.534·21-s + 1.04·22-s + 1.34·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s − 0.462·28-s − 1.73·29-s − 1.15·31-s − 0.176·32-s − 0.852·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.406325034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.406325034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 + 9.34T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 + 0.449T + 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 + 2.89T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 1.10T + 53T^{2} \) |
| 59 | \( 1 - 5.79T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 2.44T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 1.55T + 79T^{2} \) |
| 83 | \( 1 - 2.89T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 + 3.79T + 97T^{2} \) |
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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
−8.950061566508893550782138446992, −8.206566729850964485830648999336, −7.46005622566974040958674872684, −6.90477558149818345161203945081, −5.79655842802968968720557683648, −5.22614767829113747166396468607, −3.53568189339376944415190682236, −3.28096150901902320787499099393, −2.09894765522805329746365977275, −0.809615981487486687396422711816,
0.809615981487486687396422711816, 2.09894765522805329746365977275, 3.28096150901902320787499099393, 3.53568189339376944415190682236, 5.22614767829113747166396468607, 5.79655842802968968720557683648, 6.90477558149818345161203945081, 7.46005622566974040958674872684, 8.206566729850964485830648999336, 8.950061566508893550782138446992
