| L(s) = 1 | − 10.3·5-s + 3.46·7-s − 55.4·13-s + 90·17-s + 116·19-s − 103.·23-s − 17·25-s + 259.·29-s − 301.·31-s − 36·35-s + 34.6·37-s − 54·41-s + 20·43-s − 394.·47-s − 331·49-s − 488.·53-s − 324·59-s + 575.·61-s + 576·65-s − 116·67-s + 1.10e3·71-s − 1.10e3·73-s − 148.·79-s + 1.15e3·83-s − 935.·85-s + 918·89-s − 191.·91-s + ⋯ |
| L(s) = 1 | − 0.929·5-s + 0.187·7-s − 1.18·13-s + 1.28·17-s + 1.40·19-s − 0.942·23-s − 0.136·25-s + 1.66·29-s − 1.74·31-s − 0.173·35-s + 0.153·37-s − 0.205·41-s + 0.0709·43-s − 1.22·47-s − 0.965·49-s − 1.26·53-s − 0.714·59-s + 1.20·61-s + 1.09·65-s − 0.211·67-s + 1.84·71-s − 1.77·73-s − 0.212·79-s + 1.52·83-s − 1.19·85-s + 1.09·89-s − 0.221·91-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
| \(L(2)\) |
\(\approx\) |
\(1.370172426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.370172426\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 10.3T + 125T^{2} \) |
| 7 | \( 1 - 3.46T + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 90T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 259.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 301.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 54T + 6.89e4T^{2} \) |
| 43 | \( 1 - 20T + 7.95e4T^{2} \) |
| 47 | \( 1 + 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 488.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 324T + 2.05e5T^{2} \) |
| 61 | \( 1 - 575.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 116T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 148.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 918T + 7.04e5T^{2} \) |
| 97 | \( 1 - 190T + 9.12e5T^{2} \) |
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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
−8.456603692256435693296168971668, −7.69928050797399009180060390460, −7.46475177973398482117660649937, −6.37009355772922835503151262003, −5.32791366303549614004120650111, −4.74490303475695995076106125433, −3.66666250479855471757686946810, −3.02424263065069476837742375124, −1.73439316377000511640944400436, −0.52321336631476145320325708580,
0.52321336631476145320325708580, 1.73439316377000511640944400436, 3.02424263065069476837742375124, 3.66666250479855471757686946810, 4.74490303475695995076106125433, 5.32791366303549614004120650111, 6.37009355772922835503151262003, 7.46475177973398482117660649937, 7.69928050797399009180060390460, 8.456603692256435693296168971668
