| L(s) = 1 | − 19.8·3-s + 59.6·7-s + 153·9-s − 252·11-s + 119.·13-s − 689.·17-s − 220·19-s − 1.18e3·21-s + 2.43e3·23-s + 1.79e3·27-s + 6.93e3·29-s − 6.75e3·31-s + 5.01e3·33-s + 1.39e4·37-s − 2.37e3·39-s − 198·41-s − 417.·43-s + 1.05e4·47-s − 1.32e4·49-s + 1.37e4·51-s + 5.82e3·53-s + 4.37e3·57-s − 2.46e4·59-s − 5.69e3·61-s + 9.13e3·63-s + 4.36e4·67-s − 4.84e4·69-s + ⋯ |
| L(s) = 1 | − 1.27·3-s + 0.460·7-s + 0.629·9-s − 0.627·11-s + 0.195·13-s − 0.578·17-s − 0.139·19-s − 0.587·21-s + 0.959·23-s + 0.472·27-s + 1.53·29-s − 1.26·31-s + 0.801·33-s + 1.67·37-s − 0.250·39-s − 0.0183·41-s − 0.0344·43-s + 0.695·47-s − 0.787·49-s + 0.739·51-s + 0.284·53-s + 0.178·57-s − 0.922·59-s − 0.196·61-s + 0.289·63-s + 1.18·67-s − 1.22·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 19.8T + 243T^{2} \) |
| 7 | \( 1 - 59.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 252T + 1.61e5T^{2} \) |
| 13 | \( 1 - 119.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 689.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 220T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 198T + 1.15e8T^{2} \) |
| 43 | \( 1 + 417.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.82e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.99e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.01e5T + 8.58e9T^{2} \) |
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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
−10.35818821787494541279214543999, −9.123310556186932417007813270567, −8.091083777066748536762112707685, −6.99548608481181573793524656547, −6.07930988280287988379847523773, −5.18607476741954390001413273813, −4.39028019376438201796000059209, −2.72879363999033578777438028677, −1.17575047012676916630540584565, 0,
1.17575047012676916630540584565, 2.72879363999033578777438028677, 4.39028019376438201796000059209, 5.18607476741954390001413273813, 6.07930988280287988379847523773, 6.99548608481181573793524656547, 8.091083777066748536762112707685, 9.123310556186932417007813270567, 10.35818821787494541279214543999
