| L(s) = 1 | − 2-s − 0.842·3-s + 4-s + 5-s + 0.842·6-s − 3.49·7-s − 8-s − 2.28·9-s − 10-s + 3.53·11-s − 0.842·12-s + 5.00·13-s + 3.49·14-s − 0.842·15-s + 16-s + 7.60·17-s + 2.28·18-s + 2.42·19-s + 20-s + 2.94·21-s − 3.53·22-s + 3.71·23-s + 0.842·24-s + 25-s − 5.00·26-s + 4.45·27-s − 3.49·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.486·3-s + 0.5·4-s + 0.447·5-s + 0.344·6-s − 1.32·7-s − 0.353·8-s − 0.763·9-s − 0.316·10-s + 1.06·11-s − 0.243·12-s + 1.38·13-s + 0.934·14-s − 0.217·15-s + 0.250·16-s + 1.84·17-s + 0.539·18-s + 0.555·19-s + 0.223·20-s + 0.642·21-s − 0.753·22-s + 0.774·23-s + 0.172·24-s + 0.200·25-s − 0.981·26-s + 0.858·27-s − 0.660·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.510322068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510322068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 0.842T + 3T^{2} \) |
| 7 | \( 1 + 3.49T + 7T^{2} \) |
| 11 | \( 1 - 3.53T + 11T^{2} \) |
| 13 | \( 1 - 5.00T + 13T^{2} \) |
| 17 | \( 1 - 7.60T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 23 | \( 1 - 3.71T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 37 | \( 1 - 7.63T + 37T^{2} \) |
| 41 | \( 1 - 1.29T + 41T^{2} \) |
| 43 | \( 1 - 6.10T + 43T^{2} \) |
| 47 | \( 1 + 1.84T + 47T^{2} \) |
| 53 | \( 1 + 5.89T + 53T^{2} \) |
| 59 | \( 1 - 1.43T + 59T^{2} \) |
| 61 | \( 1 + 0.677T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 3.79T + 83T^{2} \) |
| 89 | \( 1 + 5.68T + 89T^{2} \) |
| 97 | \( 1 + 19.1T + 97T^{2} \) |
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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
−7.78957396263714871981229769803, −6.70029692133797155082980798556, −6.46055988117906385518742895883, −5.82317765053559738305825363158, −5.30547774178361623032968042353, −3.97114132432287923464874474046, −3.24559010578878375448346495319, −2.75017855089729142635295534388, −1.24897463585319302123532805514, −0.799389987061897996982084649734,
0.799389987061897996982084649734, 1.24897463585319302123532805514, 2.75017855089729142635295534388, 3.24559010578878375448346495319, 3.97114132432287923464874474046, 5.30547774178361623032968042353, 5.82317765053559738305825363158, 6.46055988117906385518742895883, 6.70029692133797155082980798556, 7.78957396263714871981229769803
