| L(s) = 1 | + 0.936·2-s − 3-s − 1.12·4-s − 0.936·6-s − 3.65·7-s − 2.92·8-s + 9-s − 5.57·11-s + 1.12·12-s − 2.92·13-s − 3.42·14-s − 0.494·16-s − 2.26·17-s + 0.936·18-s − 2.64·19-s + 3.65·21-s − 5.22·22-s + 7.72·23-s + 2.92·24-s − 2.74·26-s − 27-s + 4.10·28-s − 7.79·29-s − 6.03·31-s + 5.38·32-s + 5.57·33-s − 2.12·34-s + ⋯ |
| L(s) = 1 | + 0.662·2-s − 0.577·3-s − 0.561·4-s − 0.382·6-s − 1.38·7-s − 1.03·8-s + 0.333·9-s − 1.68·11-s + 0.324·12-s − 0.812·13-s − 0.915·14-s − 0.123·16-s − 0.549·17-s + 0.220·18-s − 0.606·19-s + 0.797·21-s − 1.11·22-s + 1.61·23-s + 0.597·24-s − 0.538·26-s − 0.192·27-s + 0.775·28-s − 1.44·29-s − 1.08·31-s + 0.952·32-s + 0.970·33-s − 0.363·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2938847445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2938847445\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 0.936T + 2T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 + 2.92T + 13T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 - 7.72T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 + 6.03T + 31T^{2} \) |
| 37 | \( 1 + 6.60T + 37T^{2} \) |
| 41 | \( 1 + 8.81T + 41T^{2} \) |
| 43 | \( 1 - 9.69T + 43T^{2} \) |
| 53 | \( 1 + 2.97T + 53T^{2} \) |
| 59 | \( 1 - 0.571T + 59T^{2} \) |
| 61 | \( 1 - 8.30T + 61T^{2} \) |
| 67 | \( 1 + 8.03T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 5.94T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 4.31T + 89T^{2} \) |
| 97 | \( 1 - 5.24T + 97T^{2} \) |
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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
−8.778414103120192872376501861764, −7.61138623799244190876430324565, −6.96227914563902465426197823572, −6.17952603768720263019051619185, −5.25566645059408595961836323023, −5.09026851201394818160602191538, −3.93483427604387528970911071609, −3.17407278834108528911483638349, −2.33013221754448927787512558479, −0.27509661287379030251131151382,
0.27509661287379030251131151382, 2.33013221754448927787512558479, 3.17407278834108528911483638349, 3.93483427604387528970911071609, 5.09026851201394818160602191538, 5.25566645059408595961836323023, 6.17952603768720263019051619185, 6.96227914563902465426197823572, 7.61138623799244190876430324565, 8.778414103120192872376501861764
