| L(s) = 1 | + 3-s − 3·5-s − 2·7-s − 2·9-s + 4·13-s − 3·15-s − 6·17-s − 8·19-s − 2·21-s − 3·23-s + 4·25-s − 5·27-s + 5·31-s + 6·35-s − 37-s + 4·39-s + 10·43-s + 6·45-s − 3·49-s − 6·51-s − 6·53-s − 8·57-s + 3·59-s + 4·61-s + 4·63-s − 12·65-s − 67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.755·7-s − 2/3·9-s + 1.10·13-s − 0.774·15-s − 1.45·17-s − 1.83·19-s − 0.436·21-s − 0.625·23-s + 4/5·25-s − 0.962·27-s + 0.898·31-s + 1.01·35-s − 0.164·37-s + 0.640·39-s + 1.52·43-s + 0.894·45-s − 3/7·49-s − 0.840·51-s − 0.824·53-s − 1.05·57-s + 0.390·59-s + 0.512·61-s + 0.503·63-s − 1.48·65-s − 0.122·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−10.82109708424629217833914870894, −9.431312841027853518235538734747, −8.453584718504860075735545131686, −8.204138017445950847285230309260, −6.82620843631036387433675479847, −6.07532856213092694592866222076, −4.33075028432463196258949288256, −3.68366753852883183133176875421, −2.47303941619863806873155456193, 0,
2.47303941619863806873155456193, 3.68366753852883183133176875421, 4.33075028432463196258949288256, 6.07532856213092694592866222076, 6.82620843631036387433675479847, 8.204138017445950847285230309260, 8.453584718504860075735545131686, 9.431312841027853518235538734747, 10.82109708424629217833914870894
