| L(s) = 1 | + 5.08·2-s − 6.19·4-s + 32.4·5-s + 49·7-s − 194.·8-s + 164.·10-s − 110.·11-s − 1.15e3·13-s + 248.·14-s − 787.·16-s + 653.·17-s + 2.01·19-s − 200.·20-s − 562.·22-s + 1.75e3·23-s − 2.07e3·25-s − 5.86e3·26-s − 303.·28-s − 7.73e3·29-s + 2.44e3·31-s + 2.20e3·32-s + 3.32e3·34-s + 1.58e3·35-s − 9.44e3·37-s + 10.2·38-s − 6.28e3·40-s − 1.41e4·41-s + ⋯ |
| L(s) = 1 | + 0.898·2-s − 0.193·4-s + 0.579·5-s + 0.377·7-s − 1.07·8-s + 0.520·10-s − 0.275·11-s − 1.89·13-s + 0.339·14-s − 0.769·16-s + 0.548·17-s + 0.00127·19-s − 0.112·20-s − 0.247·22-s + 0.691·23-s − 0.663·25-s − 1.70·26-s − 0.0731·28-s − 1.70·29-s + 0.457·31-s + 0.381·32-s + 0.492·34-s + 0.219·35-s − 1.13·37-s + 0.00114·38-s − 0.621·40-s − 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 2 | \( 1 - 5.08T + 32T^{2} \) |
| 5 | \( 1 - 32.4T + 3.12e3T^{2} \) |
| 11 | \( 1 + 110.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 653.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.01T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.75e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.44e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.41e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.36e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.90e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.67e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.15e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.46e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.29e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.73e4T + 8.58e9T^{2} \) |
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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
−11.50367808138215590211595733454, −10.04267235828709264827271725224, −9.393126339683408954115280192159, −8.057355015772375267570029939429, −6.82707278752020846978880437527, −5.40192097185005235740044410513, −4.92690395975296579515904849225, −3.43273719466104294079347920003, −2.08810522300848518511357460868, 0,
2.08810522300848518511357460868, 3.43273719466104294079347920003, 4.92690395975296579515904849225, 5.40192097185005235740044410513, 6.82707278752020846978880437527, 8.057355015772375267570029939429, 9.393126339683408954115280192159, 10.04267235828709264827271725224, 11.50367808138215590211595733454
