| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 4·11-s − 13-s − 15-s + 8·17-s + 6·19-s − 2·21-s − 6·23-s + 25-s + 27-s − 4·29-s − 4·33-s + 2·35-s − 2·37-s − 39-s − 2·41-s + 4·43-s − 45-s − 3·49-s + 8·51-s − 10·53-s + 4·55-s + 6·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.258·15-s + 1.94·17-s + 1.37·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 0.696·33-s + 0.338·35-s − 0.328·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s − 3/7·49-s + 1.12·51-s − 1.37·53-s + 0.539·55-s + 0.794·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.001879854227249847712710170108, −7.76948487871725340041744632337, −7.09699921362048702400785248351, −5.88823502695794701483978541538, −5.35993903747378895770836082484, −4.28385424811650436847502480563, −3.22048752563716810372968934226, −2.96849062059581912835324119736, −1.52256173769279382883262599647, 0,
1.52256173769279382883262599647, 2.96849062059581912835324119736, 3.22048752563716810372968934226, 4.28385424811650436847502480563, 5.35993903747378895770836082484, 5.88823502695794701483978541538, 7.09699921362048702400785248351, 7.76948487871725340041744632337, 8.001879854227249847712710170108
