| L(s) = 1 | − 1.41·2-s + 3-s − 0.00162·4-s + 5-s − 1.41·6-s − 1.83·7-s + 2.82·8-s + 9-s − 1.41·10-s + 4.77·11-s − 0.00162·12-s + 0.778·13-s + 2.59·14-s + 15-s − 3.99·16-s + 7.82·17-s − 1.41·18-s + 5.72·19-s − 0.00162·20-s − 1.83·21-s − 6.74·22-s + 7.45·23-s + 2.82·24-s + 25-s − 1.09·26-s + 27-s + 0.00298·28-s + ⋯ |
| L(s) = 1 | − 0.999·2-s + 0.577·3-s − 0.000813·4-s + 0.447·5-s − 0.577·6-s − 0.693·7-s + 1.00·8-s + 0.333·9-s − 0.447·10-s + 1.43·11-s − 0.000469·12-s + 0.215·13-s + 0.692·14-s + 0.258·15-s − 0.999·16-s + 1.89·17-s − 0.333·18-s + 1.31·19-s − 0.000363·20-s − 0.400·21-s − 1.43·22-s + 1.55·23-s + 0.577·24-s + 0.200·25-s − 0.215·26-s + 0.192·27-s + 0.000563·28-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
| \(L(1)\) |
\(\approx\) |
\(1.968045221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.968045221\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.41T + 2T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 - 0.778T + 13T^{2} \) |
| 17 | \( 1 - 7.82T + 17T^{2} \) |
| 19 | \( 1 - 5.72T + 19T^{2} \) |
| 23 | \( 1 - 7.45T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 0.923T + 31T^{2} \) |
| 37 | \( 1 - 2.81T + 37T^{2} \) |
| 41 | \( 1 + 1.14T + 41T^{2} \) |
| 43 | \( 1 + 2.94T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 8.82T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 0.706T + 67T^{2} \) |
| 71 | \( 1 + 5.40T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 2.65T + 79T^{2} \) |
| 83 | \( 1 - 8.65T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 6.38T + 97T^{2} \) |
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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
−8.260407260170262624740240002894, −7.45229588381815506708114633870, −6.92954645274475588987401841601, −6.14110747247214266358491912153, −5.22000242044223131678211468711, −4.37286419521347909232271829592, −3.38737510425829847769403010291, −2.86989205394971990271691101739, −1.26426490695216847480836822407, −1.09359457053326920340038227335,
1.09359457053326920340038227335, 1.26426490695216847480836822407, 2.86989205394971990271691101739, 3.38737510425829847769403010291, 4.37286419521347909232271829592, 5.22000242044223131678211468711, 6.14110747247214266358491912153, 6.92954645274475588987401841601, 7.45229588381815506708114633870, 8.260407260170262624740240002894
