| L(s) = 1 | + 3-s + 7-s − 2·9-s − 6·11-s − 3·13-s + 21-s + 23-s − 5·25-s − 5·27-s − 3·29-s − 7·31-s − 6·33-s + 8·37-s − 3·39-s − 11·41-s + 4·43-s + 47-s + 49-s + 4·53-s + 12·59-s − 6·61-s − 2·63-s + 12·67-s + 69-s − 5·71-s + 15·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 2/3·9-s − 1.80·11-s − 0.832·13-s + 0.218·21-s + 0.208·23-s − 25-s − 0.962·27-s − 0.557·29-s − 1.25·31-s − 1.04·33-s + 1.31·37-s − 0.480·39-s − 1.71·41-s + 0.609·43-s + 0.145·47-s + 1/7·49-s + 0.549·53-s + 1.56·59-s − 0.768·61-s − 0.251·63-s + 1.46·67-s + 0.120·69-s − 0.593·71-s + 1.75·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.277328109752433488354626980934, −8.221880788047053290459578459124, −7.87080963864437179411698957603, −7.02436029292618715067670405103, −5.59088807268992571534451423947, −5.22669686499644929925271513897, −3.94369158891702944415794251560, −2.79977867691787133175013368948, −2.12170711395311612311886795887, 0,
2.12170711395311612311886795887, 2.79977867691787133175013368948, 3.94369158891702944415794251560, 5.22669686499644929925271513897, 5.59088807268992571534451423947, 7.02436029292618715067670405103, 7.87080963864437179411698957603, 8.221880788047053290459578459124, 9.277328109752433488354626980934
