| L(s) = 1 | + 3·3-s − 4·5-s + 7·7-s + 9·9-s − 62·11-s − 62·13-s − 12·15-s + 84·17-s − 100·19-s + 21·21-s + 42·23-s − 109·25-s + 27·27-s − 10·29-s + 48·31-s − 186·33-s − 28·35-s − 246·37-s − 186·39-s − 248·41-s − 68·43-s − 36·45-s − 324·47-s + 49·49-s + 252·51-s + 258·53-s + 248·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.357·5-s + 0.377·7-s + 1/3·9-s − 1.69·11-s − 1.32·13-s − 0.206·15-s + 1.19·17-s − 1.20·19-s + 0.218·21-s + 0.380·23-s − 0.871·25-s + 0.192·27-s − 0.0640·29-s + 0.278·31-s − 0.981·33-s − 0.135·35-s − 1.09·37-s − 0.763·39-s − 0.944·41-s − 0.241·43-s − 0.119·45-s − 1.00·47-s + 1/7·49-s + 0.691·51-s + 0.668·53-s + 0.608·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 62 T + p^{3} T^{2} \) |
| 13 | \( 1 + 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 10 T + p^{3} T^{2} \) |
| 31 | \( 1 - 48 T + p^{3} T^{2} \) |
| 37 | \( 1 + 246 T + p^{3} T^{2} \) |
| 41 | \( 1 + 248 T + p^{3} T^{2} \) |
| 43 | \( 1 + 68 T + p^{3} T^{2} \) |
| 47 | \( 1 + 324 T + p^{3} T^{2} \) |
| 53 | \( 1 - 258 T + p^{3} T^{2} \) |
| 59 | \( 1 + 120 T + p^{3} T^{2} \) |
| 61 | \( 1 - 622 T + p^{3} T^{2} \) |
| 67 | \( 1 + 904 T + p^{3} T^{2} \) |
| 71 | \( 1 - 678 T + p^{3} T^{2} \) |
| 73 | \( 1 + 642 T + p^{3} T^{2} \) |
| 79 | \( 1 + 740 T + p^{3} T^{2} \) |
| 83 | \( 1 + 468 T + p^{3} T^{2} \) |
| 89 | \( 1 - 200 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1266 T + p^{3} T^{2} \) |
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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
−10.40469155020550031416800878880, −9.945034996028183442615143786379, −8.548176199361657625102650802274, −7.86802113558466716574309090261, −7.13105724560995435704799670344, −5.50445383329104497091345565619, −4.61471066714857130936067952631, −3.18517497809221748474725177403, −2.07939198203360361208134606541, 0,
2.07939198203360361208134606541, 3.18517497809221748474725177403, 4.61471066714857130936067952631, 5.50445383329104497091345565619, 7.13105724560995435704799670344, 7.86802113558466716574309090261, 8.548176199361657625102650802274, 9.945034996028183442615143786379, 10.40469155020550031416800878880
