| L(s) = 1 | + 1.61·3-s + 5-s + 3.85·7-s − 0.381·9-s − 1.23·11-s − 6.09·13-s + 1.61·15-s − 1.38·17-s − 7.23·19-s + 6.23·21-s − 0.854·23-s + 25-s − 5.47·27-s − 29-s + 0.618·31-s − 2.00·33-s + 3.85·35-s − 4.76·37-s − 9.85·39-s + 9.70·41-s + 5.38·43-s − 0.381·45-s − 8·47-s + 7.85·49-s − 2.23·51-s + 6.32·53-s − 1.23·55-s + ⋯ |
| L(s) = 1 | + 0.934·3-s + 0.447·5-s + 1.45·7-s − 0.127·9-s − 0.372·11-s − 1.68·13-s + 0.417·15-s − 0.335·17-s − 1.66·19-s + 1.36·21-s − 0.178·23-s + 0.200·25-s − 1.05·27-s − 0.185·29-s + 0.111·31-s − 0.348·33-s + 0.651·35-s − 0.783·37-s − 1.57·39-s + 1.51·41-s + 0.820·43-s − 0.0569·45-s − 1.16·47-s + 1.12·49-s − 0.313·51-s + 0.868·53-s − 0.166·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 + 6.09T + 13T^{2} \) |
| 17 | \( 1 + 1.38T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 0.854T + 23T^{2} \) |
| 31 | \( 1 - 0.618T + 31T^{2} \) |
| 37 | \( 1 + 4.76T + 37T^{2} \) |
| 41 | \( 1 - 9.70T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 + 6.47T + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 2.29T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−7.55206064723423335593254502470, −6.89214496612914423220694980347, −5.89588607269668331921111960380, −5.20932837965744016296418115028, −4.57100548365004783540512273773, −3.94263195039716453567812476071, −2.58686133776375308009105927657, −2.40332193047745199439211773288, −1.59483722820858025670703466595, 0,
1.59483722820858025670703466595, 2.40332193047745199439211773288, 2.58686133776375308009105927657, 3.94263195039716453567812476071, 4.57100548365004783540512273773, 5.20932837965744016296418115028, 5.89588607269668331921111960380, 6.89214496612914423220694980347, 7.55206064723423335593254502470
