| L(s) = 1 | + 1.11·2-s − 0.753·4-s + 5-s − 1.16·7-s − 3.07·8-s + 1.11·10-s + 2.19·11-s + 1.10·13-s − 1.30·14-s − 1.92·16-s − 7.78·17-s + 4.75·19-s − 0.753·20-s + 2.44·22-s − 2.26·23-s + 25-s + 1.22·26-s + 0.879·28-s + 6.01·29-s − 1.58·31-s + 3.99·32-s − 8.68·34-s − 1.16·35-s − 7.29·37-s + 5.30·38-s − 3.07·40-s + 2.30·41-s + ⋯ |
| L(s) = 1 | + 0.789·2-s − 0.376·4-s + 0.447·5-s − 0.441·7-s − 1.08·8-s + 0.353·10-s + 0.660·11-s + 0.305·13-s − 0.348·14-s − 0.481·16-s − 1.88·17-s + 1.08·19-s − 0.168·20-s + 0.521·22-s − 0.473·23-s + 0.200·25-s + 0.241·26-s + 0.166·28-s + 1.11·29-s − 0.284·31-s + 0.706·32-s − 1.48·34-s − 0.197·35-s − 1.19·37-s + 0.860·38-s − 0.486·40-s + 0.360·41-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 | \( 1 - 1.11T + 2T^{2} \) |
| 7 | \( 1 + 1.16T + 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 - 1.10T + 13T^{2} \) |
| 17 | \( 1 + 7.78T + 17T^{2} \) |
| 19 | \( 1 - 4.75T + 19T^{2} \) |
| 23 | \( 1 + 2.26T + 23T^{2} \) |
| 29 | \( 1 - 6.01T + 29T^{2} \) |
| 31 | \( 1 + 1.58T + 31T^{2} \) |
| 37 | \( 1 + 7.29T + 37T^{2} \) |
| 41 | \( 1 - 2.30T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 - 0.949T + 47T^{2} \) |
| 53 | \( 1 + 8.03T + 53T^{2} \) |
| 59 | \( 1 + 3.82T + 59T^{2} \) |
| 61 | \( 1 + 5.72T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 2.59T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.60T + 83T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
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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
−8.216860420941867203403418920126, −7.06883487658277976193197346672, −6.38660131586960686664033885595, −5.91097552334956955894040600739, −4.91717407377683495665100466755, −4.38366902062929512246306769110, −3.48495091435878125806561972854, −2.76046409449729015729018324750, −1.52985507192588514682833156098, 0,
1.52985507192588514682833156098, 2.76046409449729015729018324750, 3.48495091435878125806561972854, 4.38366902062929512246306769110, 4.91717407377683495665100466755, 5.91097552334956955894040600739, 6.38660131586960686664033885595, 7.06883487658277976193197346672, 8.216860420941867203403418920126
