| L(s) = 1 | − 3-s − 2·4-s − 5-s + 7-s + 9-s + 11-s + 2·12-s + 15-s + 4·16-s + 3·17-s − 3·19-s + 2·20-s − 21-s − 23-s + 25-s − 27-s − 2·28-s − 7·29-s + 6·31-s − 33-s − 35-s − 2·36-s − 8·37-s − 2·41-s − 5·43-s − 2·44-s − 45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.577·12-s + 0.258·15-s + 16-s + 0.727·17-s − 0.688·19-s + 0.447·20-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s − 1.29·29-s + 1.07·31-s − 0.174·33-s − 0.169·35-s − 1/3·36-s − 1.31·37-s − 0.312·41-s − 0.762·43-s − 0.301·44-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−9.430337462578200218062202779039, −8.496053516320620371043593499931, −7.899830020151268989852881692339, −6.88447028715167810681872577779, −5.84009967093002728684369364895, −5.01570492177079781617268790956, −4.24991801630843389574934371777, −3.35399164298404557991082474166, −1.49628319683318589226316651232, 0,
1.49628319683318589226316651232, 3.35399164298404557991082474166, 4.24991801630843389574934371777, 5.01570492177079781617268790956, 5.84009967093002728684369364895, 6.88447028715167810681872577779, 7.899830020151268989852881692339, 8.496053516320620371043593499931, 9.430337462578200218062202779039
