| L(s) = 1 | + 9.91·2-s − 24.6·3-s + 66.3·4-s − 25·5-s − 244.·6-s − 60.7·7-s + 341.·8-s + 367.·9-s − 247.·10-s + 121·11-s − 1.63e3·12-s + 1.13e3·13-s − 602.·14-s + 617.·15-s + 1.25e3·16-s − 978.·17-s + 3.64e3·18-s − 361·19-s − 1.65e3·20-s + 1.50e3·21-s + 1.20e3·22-s − 1.88e3·23-s − 8.42e3·24-s + 625·25-s + 1.12e4·26-s − 3.06e3·27-s − 4.03e3·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 1.58·3-s + 2.07·4-s − 0.447·5-s − 2.77·6-s − 0.468·7-s + 1.88·8-s + 1.51·9-s − 0.784·10-s + 0.301·11-s − 3.28·12-s + 1.86·13-s − 0.821·14-s + 0.708·15-s + 1.22·16-s − 0.820·17-s + 2.64·18-s − 0.229·19-s − 0.927·20-s + 0.742·21-s + 0.528·22-s − 0.742·23-s − 2.98·24-s + 0.200·25-s + 3.26·26-s − 0.808·27-s − 0.972·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.278371100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.278371100\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 9.91T + 32T^{2} \) |
| 3 | \( 1 + 24.6T + 243T^{2} \) |
| 7 | \( 1 + 60.7T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.13e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 978.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.88e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.42e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.89e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.67e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.61e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.90e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.25e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.04e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.41e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.24e5T + 8.58e9T^{2} \) |
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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
−9.364330244262053846748892595618, −8.032853237786781151208921270178, −6.75157913575331295558724429379, −6.37442883813208311227206304867, −5.78553511466162106735520418746, −4.89916615494521202910637033373, −4.04224762931295353288839347974, −3.48082958926991322322920493939, −1.92742178848272444504482374027, −0.64416172121394534133341286812,
0.64416172121394534133341286812, 1.92742178848272444504482374027, 3.48082958926991322322920493939, 4.04224762931295353288839347974, 4.89916615494521202910637033373, 5.78553511466162106735520418746, 6.37442883813208311227206304867, 6.75157913575331295558724429379, 8.032853237786781151208921270178, 9.364330244262053846748892595618
