| L(s) = 1 | − 9.88·3-s + 2.64·5-s + 3.30·7-s + 70.6·9-s − 28.3·11-s + 85.7·13-s − 26.0·15-s − 111.·17-s + 99.8·19-s − 32.6·21-s − 110.·23-s − 118.·25-s − 430.·27-s − 29·29-s − 60.0·31-s + 280.·33-s + 8.73·35-s + 119.·37-s − 847.·39-s − 285.·41-s − 249.·43-s + 186.·45-s − 226.·47-s − 332.·49-s + 1.09e3·51-s + 387.·53-s − 74.8·55-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 0.236·5-s + 0.178·7-s + 2.61·9-s − 0.776·11-s + 1.82·13-s − 0.449·15-s − 1.58·17-s + 1.20·19-s − 0.339·21-s − 0.998·23-s − 0.944·25-s − 3.07·27-s − 0.185·29-s − 0.347·31-s + 1.47·33-s + 0.0421·35-s + 0.530·37-s − 3.47·39-s − 1.08·41-s − 0.885·43-s + 0.617·45-s − 0.702·47-s − 0.968·49-s + 3.01·51-s + 1.00·53-s − 0.183·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 9.88T + 27T^{2} \) |
| 5 | \( 1 - 2.64T + 125T^{2} \) |
| 7 | \( 1 - 3.30T + 343T^{2} \) |
| 11 | \( 1 + 28.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 85.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 99.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 110.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 60.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 119.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 285.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 249.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 226.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 387.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 164.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 188.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 903.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 820.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 394.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 605.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 504.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 930.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 12.2T + 9.12e5T^{2} \) |
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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
−11.29071309355861896904556259538, −10.60806905006153782087300055999, −9.657091926311748526696958123655, −8.161500979751266406393129243111, −6.83080264589871614140044073639, −6.00014507737750860422168568334, −5.20966092793844270827605557014, −3.99702968586669491013997372259, −1.56073988704642362253496501383, 0,
1.56073988704642362253496501383, 3.99702968586669491013997372259, 5.20966092793844270827605557014, 6.00014507737750860422168568334, 6.83080264589871614140044073639, 8.161500979751266406393129243111, 9.657091926311748526696958123655, 10.60806905006153782087300055999, 11.29071309355861896904556259538
