| L(s) = 1 | − 6.65·3-s + 5·5-s − 7·7-s + 17.3·9-s − 38.2·11-s − 19.3·13-s − 33.2·15-s − 87.2·17-s + 44.2·19-s + 46.5·21-s + 218.·23-s + 25·25-s + 64.4·27-s + 46.9·29-s + 194.·31-s + 254.·33-s − 35·35-s − 366.·37-s + 128.·39-s − 339.·41-s + 226.·43-s + 86.5·45-s + 11.6·47-s + 49·49-s + 580.·51-s + 209.·53-s − 191.·55-s + ⋯ |
| L(s) = 1 | − 1.28·3-s + 0.447·5-s − 0.377·7-s + 0.641·9-s − 1.04·11-s − 0.412·13-s − 0.572·15-s − 1.24·17-s + 0.534·19-s + 0.484·21-s + 1.97·23-s + 0.200·25-s + 0.459·27-s + 0.300·29-s + 1.12·31-s + 1.34·33-s − 0.169·35-s − 1.63·37-s + 0.528·39-s − 1.29·41-s + 0.802·43-s + 0.286·45-s + 0.0362·47-s + 0.142·49-s + 1.59·51-s + 0.541·53-s − 0.468·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 + 6.65T + 27T^{2} \) |
| 11 | \( 1 + 38.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 218.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 46.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 366.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 339.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 226.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 11.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 209.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 616T + 2.05e5T^{2} \) |
| 61 | \( 1 + 320.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 14.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 952T + 3.57e5T^{2} \) |
| 73 | \( 1 - 824.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 156.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 170.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−8.394276931229722136317906976252, −7.12448867655729755119902880628, −6.76950186469420237058774765478, −5.85985201329594277154502253863, −5.09599798597243283376818147647, −4.72066855300225441516084917966, −3.21178269050764190163728250258, −2.33319207470437124870764521727, −0.937758228315355352230454524923, 0,
0.937758228315355352230454524923, 2.33319207470437124870764521727, 3.21178269050764190163728250258, 4.72066855300225441516084917966, 5.09599798597243283376818147647, 5.85985201329594277154502253863, 6.76950186469420237058774765478, 7.12448867655729755119902880628, 8.394276931229722136317906976252
