| L(s) = 1 | − 2.58·2-s + 4.67·4-s − 5-s + 1.54·7-s − 6.92·8-s + 2.58·10-s − 3.66·11-s − 2.74·13-s − 3.98·14-s + 8.54·16-s + 6.35·17-s + 0.0289·19-s − 4.67·20-s + 9.46·22-s − 6.04·23-s + 25-s + 7.08·26-s + 7.20·28-s − 0.507·29-s − 0.766·31-s − 8.22·32-s − 16.4·34-s − 1.54·35-s + 4.38·37-s − 0.0747·38-s + 6.92·40-s + 1.18·41-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.33·4-s − 0.447·5-s + 0.582·7-s − 2.44·8-s + 0.817·10-s − 1.10·11-s − 0.760·13-s − 1.06·14-s + 2.13·16-s + 1.54·17-s + 0.00663·19-s − 1.04·20-s + 2.01·22-s − 1.25·23-s + 0.200·25-s + 1.38·26-s + 1.36·28-s − 0.0943·29-s − 0.137·31-s − 1.45·32-s − 2.81·34-s − 0.260·35-s + 0.721·37-s − 0.0121·38-s + 1.09·40-s + 0.185·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 11 | \( 1 + 3.66T + 11T^{2} \) |
| 13 | \( 1 + 2.74T + 13T^{2} \) |
| 17 | \( 1 - 6.35T + 17T^{2} \) |
| 19 | \( 1 - 0.0289T + 19T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 + 0.507T + 29T^{2} \) |
| 31 | \( 1 + 0.766T + 31T^{2} \) |
| 37 | \( 1 - 4.38T + 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 + 2.39T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 6.72T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 + 5.47T + 79T^{2} \) |
| 83 | \( 1 + 6.74T + 83T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.069457237973776762690043561732, −7.56890170611308560414487325649, −7.22691768267202696104325036385, −5.99460509356310353693031715113, −5.35997669706135617794404374690, −4.18561588737019104178637199721, −2.93416326623098581351512358966, −2.21878590483203500823952128023, −1.12030135301705863191293081514, 0,
1.12030135301705863191293081514, 2.21878590483203500823952128023, 2.93416326623098581351512358966, 4.18561588737019104178637199721, 5.35997669706135617794404374690, 5.99460509356310353693031715113, 7.22691768267202696104325036385, 7.56890170611308560414487325649, 8.069457237973776762690043561732
