| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 4·13-s − 15-s − 17-s − 21-s + 23-s + 25-s − 27-s − 7·29-s − 7·31-s + 35-s − 3·37-s + 4·39-s − 5·41-s + 12·43-s + 45-s − 6·47-s − 6·49-s + 51-s + 3·53-s + 7·59-s + 2·61-s + 63-s − 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.258·15-s − 0.242·17-s − 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.29·29-s − 1.25·31-s + 0.169·35-s − 0.493·37-s + 0.640·39-s − 0.780·41-s + 1.82·43-s + 0.149·45-s − 0.875·47-s − 6/7·49-s + 0.140·51-s + 0.412·53-s + 0.911·59-s + 0.256·61-s + 0.125·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.488446451034819736442803240922, −7.43528475342911113043891016232, −7.05459856331129357303821165008, −5.99550051910892169798928549074, −5.34768266143787433439439575300, −4.68768793196728848930348918446, −3.69174505077139847729500931270, −2.45678734659435363805920412584, −1.55208876733483474880543350638, 0,
1.55208876733483474880543350638, 2.45678734659435363805920412584, 3.69174505077139847729500931270, 4.68768793196728848930348918446, 5.34768266143787433439439575300, 5.99550051910892169798928549074, 7.05459856331129357303821165008, 7.43528475342911113043891016232, 8.488446451034819736442803240922
