| L(s) = 1 | − 8.64·5-s − 4.98·7-s − 31.2·11-s + 46.5·13-s + 29.2·17-s + 87.1·19-s − 38.6·23-s − 50.2·25-s + 131.·29-s + 137.·31-s + 43.0·35-s + 51.9·37-s − 226.·41-s − 16.7·43-s + 110.·47-s − 318.·49-s + 142.·53-s + 270.·55-s − 547.·59-s + 9.17·61-s − 402.·65-s − 22.1·67-s − 1.16e3·71-s + 317.·73-s + 155.·77-s − 958.·79-s − 207.·83-s + ⋯ |
| L(s) = 1 | − 0.773·5-s − 0.269·7-s − 0.856·11-s + 0.993·13-s + 0.416·17-s + 1.05·19-s − 0.350·23-s − 0.401·25-s + 0.839·29-s + 0.795·31-s + 0.208·35-s + 0.230·37-s − 0.862·41-s − 0.0593·43-s + 0.341·47-s − 0.927·49-s + 0.368·53-s + 0.662·55-s − 1.20·59-s + 0.0192·61-s − 0.768·65-s − 0.0403·67-s − 1.94·71-s + 0.508·73-s + 0.230·77-s − 1.36·79-s − 0.273·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8.64T + 125T^{2} \) |
| 7 | \( 1 + 4.98T + 343T^{2} \) |
| 11 | \( 1 + 31.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 38.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 131.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 137.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 51.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 226.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 16.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 142.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 547.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 9.17T + 2.26e5T^{2} \) |
| 67 | \( 1 + 22.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 317.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 958.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 207.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 97.3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.21e3T + 9.12e5T^{2} \) |
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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
−8.908947255209451755568901722367, −8.066330387742725544778552939993, −7.56649638373872485199248230935, −6.48589540956258302582289408187, −5.62407828629723395306772611233, −4.62660159865972696539602193531, −3.61007776157383569494315271122, −2.82696078355623616342496341354, −1.26366655880377889535661353911, 0,
1.26366655880377889535661353911, 2.82696078355623616342496341354, 3.61007776157383569494315271122, 4.62660159865972696539602193531, 5.62407828629723395306772611233, 6.48589540956258302582289408187, 7.56649638373872485199248230935, 8.066330387742725544778552939993, 8.908947255209451755568901722367
