| L(s) = 1 | + 2.03·2-s + 2.12·4-s + 2.52·5-s − 2.32·7-s + 0.258·8-s + 5.13·10-s + 3.16·11-s + 0.552·13-s − 4.73·14-s − 3.72·16-s + 4.68·17-s + 5.06·19-s + 5.37·20-s + 6.43·22-s + 23-s + 1.38·25-s + 1.12·26-s − 4.95·28-s + 29-s − 5.94·31-s − 8.09·32-s + 9.51·34-s − 5.88·35-s + 5.66·37-s + 10.2·38-s + 0.653·40-s + 7.88·41-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.06·4-s + 1.13·5-s − 0.880·7-s + 0.0913·8-s + 1.62·10-s + 0.954·11-s + 0.153·13-s − 1.26·14-s − 0.932·16-s + 1.13·17-s + 1.16·19-s + 1.20·20-s + 1.37·22-s + 0.208·23-s + 0.277·25-s + 0.220·26-s − 0.936·28-s + 0.185·29-s − 1.06·31-s − 1.43·32-s + 1.63·34-s − 0.994·35-s + 0.930·37-s + 1.67·38-s + 0.103·40-s + 1.23·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.534732857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.534732857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.03T + 2T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 - 0.552T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 - 5.06T + 19T^{2} \) |
| 31 | \( 1 + 5.94T + 31T^{2} \) |
| 37 | \( 1 - 5.66T + 37T^{2} \) |
| 41 | \( 1 - 7.88T + 41T^{2} \) |
| 43 | \( 1 - 0.513T + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 + 2.35T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 4.39T + 67T^{2} \) |
| 71 | \( 1 - 5.37T + 71T^{2} \) |
| 73 | \( 1 + 2.76T + 73T^{2} \) |
| 79 | \( 1 - 4.88T + 79T^{2} \) |
| 83 | \( 1 - 7.48T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 4.96T + 97T^{2} \) |
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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
−7.86011653932882822653534781063, −6.96937208314223679499551583874, −6.41038360708363641767036963708, −5.69389651592055980184027092713, −5.48009077997716427873423741133, −4.42271786443339764327579020386, −3.59586956232959306014903840630, −3.08816572998014020803510593327, −2.17082061899835762244847208868, −1.04566614850452164214363716816,
1.04566614850452164214363716816, 2.17082061899835762244847208868, 3.08816572998014020803510593327, 3.59586956232959306014903840630, 4.42271786443339764327579020386, 5.48009077997716427873423741133, 5.69389651592055980184027092713, 6.41038360708363641767036963708, 6.96937208314223679499551583874, 7.86011653932882822653534781063
