| L(s) = 1 | + 2-s + 2.52·3-s + 4-s + 0.510·5-s + 2.52·6-s − 4.64·7-s + 8-s + 3.37·9-s + 0.510·10-s + 6.61·11-s + 2.52·12-s + 2.06·13-s − 4.64·14-s + 1.28·15-s + 16-s + 4.98·17-s + 3.37·18-s + 1.02·19-s + 0.510·20-s − 11.7·21-s + 6.61·22-s − 4.53·23-s + 2.52·24-s − 4.73·25-s + 2.06·26-s + 0.944·27-s − 4.64·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.45·3-s + 0.5·4-s + 0.228·5-s + 1.03·6-s − 1.75·7-s + 0.353·8-s + 1.12·9-s + 0.161·10-s + 1.99·11-s + 0.728·12-s + 0.572·13-s − 1.24·14-s + 0.332·15-s + 0.250·16-s + 1.20·17-s + 0.795·18-s + 0.234·19-s + 0.114·20-s − 2.55·21-s + 1.40·22-s − 0.945·23-s + 0.515·24-s − 0.947·25-s + 0.404·26-s + 0.181·27-s − 0.877·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.259390883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.259390883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 - 2.52T + 3T^{2} \) |
| 5 | \( 1 - 0.510T + 5T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 11 | \( 1 - 6.61T + 11T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 - 4.98T + 17T^{2} \) |
| 19 | \( 1 - 1.02T + 19T^{2} \) |
| 23 | \( 1 + 4.53T + 23T^{2} \) |
| 29 | \( 1 - 0.845T + 29T^{2} \) |
| 31 | \( 1 + 2.45T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 - 8.60T + 43T^{2} \) |
| 47 | \( 1 - 6.96T + 47T^{2} \) |
| 53 | \( 1 - 0.208T + 53T^{2} \) |
| 59 | \( 1 - 0.179T + 59T^{2} \) |
| 61 | \( 1 + 0.198T + 61T^{2} \) |
| 67 | \( 1 - 0.449T + 67T^{2} \) |
| 71 | \( 1 - 9.86T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 + 4.59T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 + 8.44T + 89T^{2} \) |
| 97 | \( 1 - 1.41T + 97T^{2} \) |
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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
−8.548211782413257662047971828464, −7.60939819226431610851315680942, −6.97642566099609424696165753692, −6.10320188826404369147277300782, −5.81928264542363820271857535945, −4.03629951912079869524992905863, −3.82310157522117197475433757479, −3.18066337374984646983148341863, −2.31126600821654024809832088587, −1.20482092171196254016622299250,
1.20482092171196254016622299250, 2.31126600821654024809832088587, 3.18066337374984646983148341863, 3.82310157522117197475433757479, 4.03629951912079869524992905863, 5.81928264542363820271857535945, 6.10320188826404369147277300782, 6.97642566099609424696165753692, 7.60939819226431610851315680942, 8.548211782413257662047971828464
