| L(s) = 1 | − 2.51·2-s − 2.89·3-s + 4.31·4-s + 5-s + 7.28·6-s − 7-s − 5.80·8-s + 5.40·9-s − 2.51·10-s − 12.5·12-s + 0.950·13-s + 2.51·14-s − 2.89·15-s + 5.96·16-s + 7.07·17-s − 13.5·18-s + 7.18·19-s + 4.31·20-s + 2.89·21-s − 8.07·23-s + 16.8·24-s + 25-s − 2.38·26-s − 6.97·27-s − 4.31·28-s + 5.89·29-s + 7.28·30-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 1.67·3-s + 2.15·4-s + 0.447·5-s + 2.97·6-s − 0.377·7-s − 2.05·8-s + 1.80·9-s − 0.794·10-s − 3.60·12-s + 0.263·13-s + 0.671·14-s − 0.748·15-s + 1.49·16-s + 1.71·17-s − 3.20·18-s + 1.64·19-s + 0.964·20-s + 0.632·21-s − 1.68·23-s + 3.43·24-s + 0.200·25-s − 0.468·26-s − 1.34·27-s − 0.814·28-s + 1.09·29-s + 1.32·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5471514082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5471514082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 + 2.89T + 3T^{2} \) |
| 13 | \( 1 - 0.950T + 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 - 7.18T + 19T^{2} \) |
| 23 | \( 1 + 8.07T + 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 - 0.725T + 31T^{2} \) |
| 37 | \( 1 - 4.45T + 37T^{2} \) |
| 41 | \( 1 - 5.00T + 41T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 4.15T + 53T^{2} \) |
| 59 | \( 1 + 1.72T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 8.66T + 71T^{2} \) |
| 73 | \( 1 - 4.74T + 73T^{2} \) |
| 79 | \( 1 - 1.73T + 79T^{2} \) |
| 83 | \( 1 - 6.45T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−8.317893529717668713406841431634, −7.67836568948449599036580960259, −7.04230271434162927737223966776, −6.21770094389343742684041639981, −5.82586847314647123192968947615, −5.07257535047134097605459657160, −3.71333999323545224672892365002, −2.47787694451926675844351881749, −1.20291759840940124961180361799, −0.72790951057237173861867295940,
0.72790951057237173861867295940, 1.20291759840940124961180361799, 2.47787694451926675844351881749, 3.71333999323545224672892365002, 5.07257535047134097605459657160, 5.82586847314647123192968947615, 6.21770094389343742684041639981, 7.04230271434162927737223966776, 7.67836568948449599036580960259, 8.317893529717668713406841431634
