| L(s) = 1 | − 0.618·2-s − 1.61·4-s + 2.23·7-s + 2.23·8-s + 4·11-s − 6.47·13-s − 1.38·14-s + 1.85·16-s + 1.23·17-s − 19-s − 2.47·22-s + 3.23·23-s + 4.00·26-s − 3.61·28-s − 7.47·29-s − 10.4·31-s − 5.61·32-s − 0.763·34-s − 2.76·37-s + 0.618·38-s + 5·41-s − 4·43-s − 6.47·44-s − 2.00·46-s + 0.472·47-s − 1.99·49-s + 10.4·52-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s + 0.845·7-s + 0.790·8-s + 1.20·11-s − 1.79·13-s − 0.369·14-s + 0.463·16-s + 0.299·17-s − 0.229·19-s − 0.527·22-s + 0.674·23-s + 0.784·26-s − 0.683·28-s − 1.38·29-s − 1.88·31-s − 0.993·32-s − 0.131·34-s − 0.454·37-s + 0.100·38-s + 0.780·41-s − 0.609·43-s − 0.975·44-s − 0.294·46-s + 0.0688·47-s − 0.285·49-s + 1.45·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 2.76T + 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 0.472T + 47T^{2} \) |
| 53 | \( 1 + 5T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 - 7.94T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 + 2.29T + 79T^{2} \) |
| 83 | \( 1 + 6.94T + 83T^{2} \) |
| 89 | \( 1 - 9.94T + 89T^{2} \) |
| 97 | \( 1 - 1.70T + 97T^{2} \) |
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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
−8.057898938355837691639521484056, −7.38086382053539253826486609194, −6.87939146538657554030191338468, −5.50760582217641699631775513088, −5.10581124589227659225998728463, −4.25012885496053050844813516868, −3.58282404942033941404110028608, −2.18257940872263039331088837329, −1.32676029829889693244601117597, 0,
1.32676029829889693244601117597, 2.18257940872263039331088837329, 3.58282404942033941404110028608, 4.25012885496053050844813516868, 5.10581124589227659225998728463, 5.50760582217641699631775513088, 6.87939146538657554030191338468, 7.38086382053539253826486609194, 8.057898938355837691639521484056
