| L(s) = 1 | + 9.15·5-s − 27.4·7-s + 20.5·11-s + 32.0·13-s + 111.·17-s − 129.·19-s + 9.16·23-s − 41.1·25-s − 41.0·29-s − 187.·31-s − 251.·35-s − 114.·37-s + 282.·41-s + 89.3·43-s + 54.6·47-s + 408.·49-s − 726.·53-s + 187.·55-s − 216.·59-s + 754.·61-s + 293.·65-s − 379.·67-s + 302.·71-s + 504.·73-s − 562.·77-s + 301.·79-s + 599.·83-s + ⋯ |
| L(s) = 1 | + 0.818·5-s − 1.48·7-s + 0.562·11-s + 0.683·13-s + 1.59·17-s − 1.56·19-s + 0.0830·23-s − 0.329·25-s − 0.262·29-s − 1.08·31-s − 1.21·35-s − 0.507·37-s + 1.07·41-s + 0.317·43-s + 0.169·47-s + 1.19·49-s − 1.88·53-s + 0.460·55-s − 0.477·59-s + 1.58·61-s + 0.559·65-s − 0.691·67-s + 0.504·71-s + 0.808·73-s − 0.832·77-s + 0.429·79-s + 0.792·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 9.15T + 125T^{2} \) |
| 7 | \( 1 + 27.4T + 343T^{2} \) |
| 11 | \( 1 - 20.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 9.16T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 89.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 54.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 726.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 754.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 379.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 504.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 301.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 599.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 277.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 765.T + 9.12e5T^{2} \) |
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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
−8.372851705259451940821405851294, −7.41232439831194670696000273754, −6.41968218021287014293133618087, −6.12791957517825065652265645149, −5.33056899450927704920521415740, −3.96037047807753391509614626585, −3.40438550102475675079778359267, −2.32086857435270799238705390177, −1.26375981333928857382622728934, 0,
1.26375981333928857382622728934, 2.32086857435270799238705390177, 3.40438550102475675079778359267, 3.96037047807753391509614626585, 5.33056899450927704920521415740, 6.12791957517825065652265645149, 6.41968218021287014293133618087, 7.41232439831194670696000273754, 8.372851705259451940821405851294
