| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 5·11-s − 4·13-s + 15-s − 3·17-s + 19-s + 21-s − 4·25-s + 27-s + 4·29-s + 2·31-s − 5·33-s + 35-s − 4·37-s − 4·39-s − 4·41-s − 5·43-s + 45-s + 7·47-s − 6·49-s − 3·51-s + 2·53-s − 5·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 1.10·13-s + 0.258·15-s − 0.727·17-s + 0.229·19-s + 0.218·21-s − 4/5·25-s + 0.192·27-s + 0.742·29-s + 0.359·31-s − 0.870·33-s + 0.169·35-s − 0.657·37-s − 0.640·39-s − 0.624·41-s − 0.762·43-s + 0.149·45-s + 1.02·47-s − 6/7·49-s − 0.420·51-s + 0.274·53-s − 0.674·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.008491284737741127609390074874, −7.66299311390565758237691526097, −6.80288736208394869637613187884, −5.87055548457730052486083795733, −4.98045618757008529895936394004, −4.54140185311433213364050286692, −3.21114450291453004016789623584, −2.50063762131778753294988518028, −1.73632279962383319477538683866, 0,
1.73632279962383319477538683866, 2.50063762131778753294988518028, 3.21114450291453004016789623584, 4.54140185311433213364050286692, 4.98045618757008529895936394004, 5.87055548457730052486083795733, 6.80288736208394869637613187884, 7.66299311390565758237691526097, 8.008491284737741127609390074874
