| L(s) = 1 | − 2.71·2-s + 5.38·4-s − 5-s + 0.704·7-s − 9.18·8-s + 2.71·10-s + 5.62·11-s − 3.49·13-s − 1.91·14-s + 14.1·16-s − 3.49·17-s − 7.97·19-s − 5.38·20-s − 15.2·22-s − 6.46·23-s + 25-s + 9.49·26-s + 3.78·28-s + 2.61·29-s − 2.43·31-s − 20.2·32-s + 9.49·34-s − 0.704·35-s − 3.96·37-s + 21.6·38-s + 9.18·40-s + 6.06·41-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.69·4-s − 0.447·5-s + 0.266·7-s − 3.24·8-s + 0.859·10-s + 1.69·11-s − 0.969·13-s − 0.511·14-s + 3.54·16-s − 0.847·17-s − 1.82·19-s − 1.20·20-s − 3.25·22-s − 1.34·23-s + 0.200·25-s + 1.86·26-s + 0.716·28-s + 0.486·29-s − 0.436·31-s − 3.57·32-s + 1.62·34-s − 0.119·35-s − 0.652·37-s + 3.51·38-s + 1.45·40-s + 0.946·41-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
| \(L(1)\) |
\(\approx\) |
\(0.5141384610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5141384610\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.71T + 2T^{2} \) |
| 7 | \( 1 - 0.704T + 7T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 + 7.97T + 19T^{2} \) |
| 23 | \( 1 + 6.46T + 23T^{2} \) |
| 29 | \( 1 - 2.61T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 + 3.96T + 37T^{2} \) |
| 41 | \( 1 - 6.06T + 41T^{2} \) |
| 43 | \( 1 - 7.29T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 + 6.05T + 53T^{2} \) |
| 59 | \( 1 - 3.69T + 59T^{2} \) |
| 61 | \( 1 - 2.26T + 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 3.71T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.416744849014948282504859426331, −8.027018161042205334970562898997, −7.09722804690072353505604886873, −6.60618937172339795842746236579, −6.01865550796300991006742936549, −4.52682457269481239701837178819, −3.71506455110081840937852803159, −2.35407224318714214050555601698, −1.81111289837489359428468094327, −0.53470603913670208307755366861,
0.53470603913670208307755366861, 1.81111289837489359428468094327, 2.35407224318714214050555601698, 3.71506455110081840937852803159, 4.52682457269481239701837178819, 6.01865550796300991006742936549, 6.60618937172339795842746236579, 7.09722804690072353505604886873, 8.027018161042205334970562898997, 8.416744849014948282504859426331
