| L(s) = 1 | − 2.48·2-s + 3-s + 4.19·4-s + 5-s − 2.48·6-s − 2.76·7-s − 5.47·8-s + 9-s − 2.48·10-s − 4.46·11-s + 4.19·12-s − 13-s + 6.87·14-s + 15-s + 5.23·16-s − 2.99·17-s − 2.48·18-s − 4.12·19-s + 4.19·20-s − 2.76·21-s + 11.1·22-s + 5.45·23-s − 5.47·24-s + 25-s + 2.48·26-s + 27-s − 11.5·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.577·3-s + 2.09·4-s + 0.447·5-s − 1.01·6-s − 1.04·7-s − 1.93·8-s + 0.333·9-s − 0.787·10-s − 1.34·11-s + 1.21·12-s − 0.277·13-s + 1.83·14-s + 0.258·15-s + 1.30·16-s − 0.727·17-s − 0.586·18-s − 0.946·19-s + 0.939·20-s − 0.602·21-s + 2.37·22-s + 1.13·23-s − 1.11·24-s + 0.200·25-s + 0.488·26-s + 0.192·27-s − 2.19·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5819900133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5819900133\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 + 4.46T + 11T^{2} \) |
| 17 | \( 1 + 2.99T + 17T^{2} \) |
| 19 | \( 1 + 4.12T + 19T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 37 | \( 1 + 0.00360T + 37T^{2} \) |
| 41 | \( 1 + 9.82T + 41T^{2} \) |
| 43 | \( 1 + 0.547T + 43T^{2} \) |
| 47 | \( 1 - 8.15T + 47T^{2} \) |
| 53 | \( 1 - 6.12T + 53T^{2} \) |
| 59 | \( 1 - 7.43T + 59T^{2} \) |
| 61 | \( 1 + 0.419T + 61T^{2} \) |
| 67 | \( 1 + 3.87T + 67T^{2} \) |
| 71 | \( 1 + 9.01T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 3.59T + 79T^{2} \) |
| 83 | \( 1 - 7.06T + 83T^{2} \) |
| 89 | \( 1 + 2.47T + 89T^{2} \) |
| 97 | \( 1 + 7.35T + 97T^{2} \) |
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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
−8.426382755371613572159585241474, −7.45490514147127777598143242916, −6.92794676714862795543307771844, −6.39289275146168011641815274142, −5.42957564842698861370345269115, −4.38385851180921973814212225823, −2.98820726225036945529946089499, −2.64339216359481680873453257227, −1.77304270591055545154323044982, −0.48430037355684994622359031680,
0.48430037355684994622359031680, 1.77304270591055545154323044982, 2.64339216359481680873453257227, 2.98820726225036945529946089499, 4.38385851180921973814212225823, 5.42957564842698861370345269115, 6.39289275146168011641815274142, 6.92794676714862795543307771844, 7.45490514147127777598143242916, 8.426382755371613572159585241474
