| L(s) = 1 | − 1.69·5-s − 17.1·7-s + 4.00·11-s + 41.0·13-s − 3.32·17-s + 108.·19-s + 142.·23-s − 122.·25-s − 295.·29-s − 239.·31-s + 29.0·35-s − 121.·37-s + 344.·41-s − 420.·43-s + 92.7·47-s − 49.1·49-s − 191.·53-s − 6.77·55-s − 661.·59-s − 359.·61-s − 69.4·65-s − 273.·67-s + 344.·71-s − 824.·73-s − 68.6·77-s + 289.·79-s + 1.32e3·83-s + ⋯ |
| L(s) = 1 | − 0.151·5-s − 0.925·7-s + 0.109·11-s + 0.875·13-s − 0.0474·17-s + 1.31·19-s + 1.29·23-s − 0.977·25-s − 1.89·29-s − 1.38·31-s + 0.140·35-s − 0.540·37-s + 1.31·41-s − 1.49·43-s + 0.287·47-s − 0.143·49-s − 0.495·53-s − 0.0166·55-s − 1.46·59-s − 0.754·61-s − 0.132·65-s − 0.499·67-s + 0.575·71-s − 1.32·73-s − 0.101·77-s + 0.411·79-s + 1.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 1.69T + 125T^{2} \) |
| 7 | \( 1 + 17.1T + 343T^{2} \) |
| 11 | \( 1 - 4.00T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 3.32T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 295.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 239.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 121.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 344.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 420.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 92.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 191.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 661.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 359.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 273.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 344.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 824.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 289.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 328.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 767.T + 9.12e5T^{2} \) |
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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
−9.405972727964585237231825344262, −9.218722640031364482414503294018, −7.84678357779523972267364813537, −7.10080978797935277254534299192, −6.08644449127345988066175931336, −5.25939492838950541430284822445, −3.81359204338220403816588593512, −3.14222434354455286557490419559, −1.51904171903100773035616475147, 0,
1.51904171903100773035616475147, 3.14222434354455286557490419559, 3.81359204338220403816588593512, 5.25939492838950541430284822445, 6.08644449127345988066175931336, 7.10080978797935277254534299192, 7.84678357779523972267364813537, 9.218722640031364482414503294018, 9.405972727964585237231825344262
