| L(s) = 1 | − 1.02·2-s − 0.945·4-s + 5-s + 2.58·7-s + 3.02·8-s − 1.02·10-s + 3.43·11-s + 3.41·13-s − 2.64·14-s − 1.21·16-s − 0.747·17-s − 5.35·19-s − 0.945·20-s − 3.53·22-s − 9.33·23-s + 25-s − 3.50·26-s − 2.43·28-s − 0.630·29-s − 10.6·31-s − 4.80·32-s + 0.767·34-s + 2.58·35-s − 8.33·37-s + 5.50·38-s + 3.02·40-s − 10.6·41-s + ⋯ |
| L(s) = 1 | − 0.726·2-s − 0.472·4-s + 0.447·5-s + 0.975·7-s + 1.06·8-s − 0.324·10-s + 1.03·11-s + 0.946·13-s − 0.708·14-s − 0.303·16-s − 0.181·17-s − 1.22·19-s − 0.211·20-s − 0.753·22-s − 1.94·23-s + 0.200·25-s − 0.687·26-s − 0.460·28-s − 0.116·29-s − 1.92·31-s − 0.848·32-s + 0.131·34-s + 0.436·35-s − 1.37·37-s + 0.892·38-s + 0.478·40-s − 1.66·41-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 | \( 1 + 1.02T + 2T^{2} \) |
| 7 | \( 1 - 2.58T + 7T^{2} \) |
| 11 | \( 1 - 3.43T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 0.747T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 + 9.33T + 23T^{2} \) |
| 29 | \( 1 + 0.630T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 8.33T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 + 0.221T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 4.16T + 59T^{2} \) |
| 61 | \( 1 - 1.13T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 0.703T + 73T^{2} \) |
| 79 | \( 1 + 6.91T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.447171953802950949481235474929, −7.53177139721715044379133583289, −6.69618487925312589601836771286, −5.88775610041190601783932725615, −5.08930352356550529549856013945, −4.16241969865406994426239573197, −3.68240126759906489047501353566, −1.79297100066757210555563801473, −1.64247532325506821289148900234, 0,
1.64247532325506821289148900234, 1.79297100066757210555563801473, 3.68240126759906489047501353566, 4.16241969865406994426239573197, 5.08930352356550529549856013945, 5.88775610041190601783932725615, 6.69618487925312589601836771286, 7.53177139721715044379133583289, 8.447171953802950949481235474929
