| L(s) = 1 | − 2.02·2-s − 1.59·3-s + 2.12·4-s − 2.38·5-s + 3.23·6-s − 2.16·7-s − 0.244·8-s − 0.464·9-s + 4.83·10-s − 0.350·11-s − 3.37·12-s − 1.38·13-s + 4.39·14-s + 3.79·15-s − 3.74·16-s − 17-s + 0.942·18-s − 7.93·19-s − 5.05·20-s + 3.45·21-s + 0.712·22-s − 7.04·23-s + 0.389·24-s + 0.677·25-s + 2.80·26-s + 5.51·27-s − 4.59·28-s + ⋯ |
| L(s) = 1 | − 1.43·2-s − 0.919·3-s + 1.06·4-s − 1.06·5-s + 1.31·6-s − 0.819·7-s − 0.0865·8-s − 0.154·9-s + 1.52·10-s − 0.105·11-s − 0.974·12-s − 0.383·13-s + 1.17·14-s + 0.979·15-s − 0.936·16-s − 0.242·17-s + 0.222·18-s − 1.81·19-s − 1.12·20-s + 0.753·21-s + 0.151·22-s − 1.46·23-s + 0.0795·24-s + 0.135·25-s + 0.549·26-s + 1.06·27-s − 0.868·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03347334462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03347334462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 2.02T + 2T^{2} \) |
| 3 | \( 1 + 1.59T + 3T^{2} \) |
| 5 | \( 1 + 2.38T + 5T^{2} \) |
| 7 | \( 1 + 2.16T + 7T^{2} \) |
| 11 | \( 1 + 0.350T + 11T^{2} \) |
| 13 | \( 1 + 1.38T + 13T^{2} \) |
| 19 | \( 1 + 7.93T + 19T^{2} \) |
| 23 | \( 1 + 7.04T + 23T^{2} \) |
| 29 | \( 1 + 1.13T + 29T^{2} \) |
| 31 | \( 1 + 5.69T + 31T^{2} \) |
| 37 | \( 1 + 1.95T + 37T^{2} \) |
| 41 | \( 1 + 7.56T + 41T^{2} \) |
| 43 | \( 1 + 8.31T + 43T^{2} \) |
| 47 | \( 1 - 8.74T + 47T^{2} \) |
| 53 | \( 1 - 7.45T + 53T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 4.06T + 79T^{2} \) |
| 83 | \( 1 - 0.755T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 0.193T + 97T^{2} \) |
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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
−10.16223361027478115355120696166, −9.031970128788359838135402889973, −8.411821949907983617260086618812, −7.60763579525346678071317041013, −6.75266474201525315632581648993, −6.03692250159333642086705480894, −4.71227820093635235121112972167, −3.68465652404015144213964207166, −2.10931883840000104780812776110, −0.17194311918149417287596471901,
0.17194311918149417287596471901, 2.10931883840000104780812776110, 3.68465652404015144213964207166, 4.71227820093635235121112972167, 6.03692250159333642086705480894, 6.75266474201525315632581648993, 7.60763579525346678071317041013, 8.411821949907983617260086618812, 9.031970128788359838135402889973, 10.16223361027478115355120696166
