| L(s) = 1 | + 9.37·2-s + 55.8·4-s − 0.277·5-s + 105.·7-s + 223.·8-s − 2.59·10-s − 147.·13-s + 984.·14-s + 307.·16-s − 1.43e3·17-s − 2.03e3·19-s − 15.4·20-s − 828.·23-s − 3.12e3·25-s − 1.38e3·26-s + 5.86e3·28-s + 4.63e3·29-s − 9.83e3·31-s − 4.27e3·32-s − 1.34e4·34-s − 29.1·35-s + 7.13e3·37-s − 1.90e4·38-s − 61.9·40-s + 1.82e4·41-s − 1.38e4·43-s − 7.76e3·46-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 1.74·4-s − 0.00495·5-s + 0.810·7-s + 1.23·8-s − 0.00821·10-s − 0.242·13-s + 1.34·14-s + 0.299·16-s − 1.20·17-s − 1.29·19-s − 0.00865·20-s − 0.326·23-s − 0.999·25-s − 0.401·26-s + 1.41·28-s + 1.02·29-s − 1.83·31-s − 0.737·32-s − 1.99·34-s − 0.00401·35-s + 0.856·37-s − 2.14·38-s − 0.00612·40-s + 1.69·41-s − 1.14·43-s − 0.541·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 9.37T + 32T^{2} \) |
| 5 | \( 1 + 0.277T + 3.12e3T^{2} \) |
| 7 | \( 1 - 105.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 147.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.43e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 828.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.82e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.08e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.49e5T + 8.58e9T^{2} \) |
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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
−8.566336900563722790498523357812, −7.70741204655303759286400657426, −6.67472491105618210030220843255, −6.06684158990375712193917929059, −5.04952876902203971211477089819, −4.44274179819487309532323432235, −3.70174532888551101061617380441, −2.44043715985697775727899793437, −1.80371116891261997279370554382, 0,
1.80371116891261997279370554382, 2.44043715985697775727899793437, 3.70174532888551101061617380441, 4.44274179819487309532323432235, 5.04952876902203971211477089819, 6.06684158990375712193917929059, 6.67472491105618210030220843255, 7.70741204655303759286400657426, 8.566336900563722790498523357812
