| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 2·7-s − 8-s + 9-s + 2·10-s + 3·11-s + 12-s − 5·13-s + 2·14-s − 2·15-s + 16-s + 7·17-s − 18-s − 5·19-s − 2·20-s − 2·21-s − 3·22-s − 4·23-s − 24-s − 25-s + 5·26-s + 27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.904·11-s + 0.288·12-s − 1.38·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s − 1.14·19-s − 0.447·20-s − 0.436·21-s − 0.639·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.980·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.864701877333940298291207314459, −9.007660079477580141510770908246, −8.184065983773167685550511616651, −7.39121605883311337973891499090, −6.77686019170865019878523434167, −5.50004206915861886604460589070, −3.99654645002136310469009194108, −3.30471830679599508220370425065, −1.90813808533815055195502247129, 0,
1.90813808533815055195502247129, 3.30471830679599508220370425065, 3.99654645002136310469009194108, 5.50004206915861886604460589070, 6.77686019170865019878523434167, 7.39121605883311337973891499090, 8.184065983773167685550511616651, 9.007660079477580141510770908246, 9.864701877333940298291207314459
