| L(s) = 1 | − 4.90·2-s + 16.0·4-s − 5·5-s − 17.0·7-s − 39.7·8-s + 24.5·10-s − 11·11-s + 61.7·13-s + 83.7·14-s + 66.1·16-s + 77.4·17-s − 98.7·19-s − 80.4·20-s + 53.9·22-s + 8.50·23-s + 25·25-s − 302.·26-s − 274.·28-s − 180.·29-s + 88.3·31-s − 7.04·32-s − 379.·34-s + 85.3·35-s + 335.·37-s + 484.·38-s + 198.·40-s − 50.9·41-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.01·4-s − 0.447·5-s − 0.921·7-s − 1.75·8-s + 0.776·10-s − 0.301·11-s + 1.31·13-s + 1.59·14-s + 1.03·16-s + 1.10·17-s − 1.19·19-s − 0.899·20-s + 0.523·22-s + 0.0771·23-s + 0.200·25-s − 2.28·26-s − 1.85·28-s − 1.15·29-s + 0.511·31-s − 0.0388·32-s − 1.91·34-s + 0.412·35-s + 1.49·37-s + 2.06·38-s + 0.784·40-s − 0.194·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.90T + 8T^{2} \) |
| 7 | \( 1 + 17.0T + 343T^{2} \) |
| 13 | \( 1 - 61.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 77.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 98.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 8.50T + 1.21e4T^{2} \) |
| 29 | \( 1 + 180.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 335.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 50.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 334.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 236.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 703.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 468.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 148.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 525.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 968.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.17e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 679.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 791.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 180.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 419.T + 9.12e5T^{2} \) |
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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
−9.927520231653808106603774298843, −9.220209561547617803812360702626, −8.333913215471960009976477286918, −7.69859775298944812237532588151, −6.65244318674516777539279058440, −5.88029633742811600698016074213, −3.93706741848610298631678728403, −2.71925493403832986955158758793, −1.20242935561992877013065083777, 0,
1.20242935561992877013065083777, 2.71925493403832986955158758793, 3.93706741848610298631678728403, 5.88029633742811600698016074213, 6.65244318674516777539279058440, 7.69859775298944812237532588151, 8.333913215471960009976477286918, 9.220209561547617803812360702626, 9.927520231653808106603774298843
