| L(s) = 1 | + 2.74·2-s + 3-s + 5.51·4-s − 5-s + 2.74·6-s + 1.51·7-s + 9.64·8-s + 9-s − 2.74·10-s + 4.31·11-s + 5.51·12-s + 2.29·13-s + 4.16·14-s − 15-s + 15.4·16-s − 3.84·17-s + 2.74·18-s + 3.75·19-s − 5.51·20-s + 1.51·21-s + 11.8·22-s − 4.27·23-s + 9.64·24-s + 25-s + 6.28·26-s + 27-s + 8.38·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 0.577·3-s + 2.75·4-s − 0.447·5-s + 1.11·6-s + 0.574·7-s + 3.40·8-s + 0.333·9-s − 0.866·10-s + 1.30·11-s + 1.59·12-s + 0.636·13-s + 1.11·14-s − 0.258·15-s + 3.85·16-s − 0.933·17-s + 0.646·18-s + 0.861·19-s − 1.23·20-s + 0.331·21-s + 2.52·22-s − 0.892·23-s + 1.96·24-s + 0.200·25-s + 1.23·26-s + 0.192·27-s + 1.58·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.22114167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.22114167\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 7 | \( 1 - 1.51T + 7T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 - 2.29T + 13T^{2} \) |
| 17 | \( 1 + 3.84T + 17T^{2} \) |
| 19 | \( 1 - 3.75T + 19T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 + 7.01T + 29T^{2} \) |
| 31 | \( 1 + 4.69T + 31T^{2} \) |
| 37 | \( 1 - 0.403T + 37T^{2} \) |
| 41 | \( 1 + 1.15T + 41T^{2} \) |
| 43 | \( 1 + 6.31T + 43T^{2} \) |
| 47 | \( 1 - 5.85T + 47T^{2} \) |
| 53 | \( 1 + 2.83T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 1.43T + 61T^{2} \) |
| 67 | \( 1 + 7.90T + 67T^{2} \) |
| 71 | \( 1 + 6.71T + 71T^{2} \) |
| 73 | \( 1 + 5.05T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 2.28T + 83T^{2} \) |
| 89 | \( 1 - 9.47T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−7.65933287605814637497954780582, −7.30667281787299051715385215593, −6.41261099491321119439651920566, −5.92981456627033398688951773701, −4.95086788010230619749876508072, −4.34485220265304078155049075023, −3.69031640805233546066494367138, −3.26134042154040902291255987575, −2.04399367192065894909666149501, −1.48826917943275010854351283655,
1.48826917943275010854351283655, 2.04399367192065894909666149501, 3.26134042154040902291255987575, 3.69031640805233546066494367138, 4.34485220265304078155049075023, 4.95086788010230619749876508072, 5.92981456627033398688951773701, 6.41261099491321119439651920566, 7.30667281787299051715385215593, 7.65933287605814637497954780582
