| L(s) = 1 | − 17.5·3-s + 11.0·5-s + 64.1·9-s − 95.0·11-s + 157.·13-s − 192.·15-s − 100.·17-s + 664.·19-s + 2.14e3·23-s − 3.00e3·25-s + 3.13e3·27-s − 3.05e3·29-s + 3.08e3·31-s + 1.66e3·33-s + 1.18e4·37-s − 2.75e3·39-s + 4.01e3·41-s − 1.03e4·43-s + 705.·45-s + 2.26e4·47-s + 1.76e3·51-s − 2.77e4·53-s − 1.04e3·55-s − 1.16e4·57-s + 2.33e4·59-s − 1.02e4·61-s + 1.73e3·65-s + ⋯ |
| L(s) = 1 | − 1.12·3-s + 0.196·5-s + 0.263·9-s − 0.236·11-s + 0.258·13-s − 0.221·15-s − 0.0843·17-s + 0.422·19-s + 0.844·23-s − 0.961·25-s + 0.827·27-s − 0.673·29-s + 0.576·31-s + 0.266·33-s + 1.42·37-s − 0.290·39-s + 0.372·41-s − 0.854·43-s + 0.0519·45-s + 1.49·47-s + 0.0948·51-s − 1.35·53-s − 0.0466·55-s − 0.474·57-s + 0.874·59-s − 0.352·61-s + 0.0508·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 17.5T + 243T^{2} \) |
| 5 | \( 1 - 11.0T + 3.12e3T^{2} \) |
| 11 | \( 1 + 95.0T + 1.61e5T^{2} \) |
| 13 | \( 1 - 157.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 100.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 664.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.14e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.18e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.26e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.77e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.55e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 313.T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.07e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.64e5T + 8.58e9T^{2} \) |
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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
−10.17786650436377731500952082562, −9.268260538529816920355622810768, −8.111429745280427116293945208394, −7.02636535032591576405382740225, −6.02109077094509991029808251216, −5.36810019963745594095922903381, −4.25629301594253340312150732976, −2.77984083889329747454762672879, −1.21261428013051388212681655538, 0,
1.21261428013051388212681655538, 2.77984083889329747454762672879, 4.25629301594253340312150732976, 5.36810019963745594095922903381, 6.02109077094509991029808251216, 7.02636535032591576405382740225, 8.111429745280427116293945208394, 9.268260538529816920355622810768, 10.17786650436377731500952082562
