L(s) = 1 | + 0.415·2-s + 3-s − 1.82·4-s + 4.32·5-s + 0.415·6-s + 7-s − 1.59·8-s + 9-s + 1.79·10-s + 3.32·11-s − 1.82·12-s + 0.415·14-s + 4.32·15-s + 2.99·16-s − 5.26·17-s + 0.415·18-s + 0.118·19-s − 7.89·20-s + 21-s + 1.38·22-s − 7.51·23-s − 1.59·24-s + 13.6·25-s + 27-s − 1.82·28-s − 2.46·29-s + 1.79·30-s + ⋯ |
L(s) = 1 | + 0.293·2-s + 0.577·3-s − 0.913·4-s + 1.93·5-s + 0.169·6-s + 0.377·7-s − 0.562·8-s + 0.333·9-s + 0.567·10-s + 1.00·11-s − 0.527·12-s + 0.111·14-s + 1.11·15-s + 0.748·16-s − 1.27·17-s + 0.0979·18-s + 0.0271·19-s − 1.76·20-s + 0.218·21-s + 0.294·22-s − 1.56·23-s − 0.324·24-s + 2.73·25-s + 0.192·27-s − 0.345·28-s − 0.456·29-s + 0.327·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.486296947\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.486296947\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.415T + 2T^{2} \) |
| 5 | \( 1 - 4.32T + 5T^{2} \) |
| 11 | \( 1 - 3.32T + 11T^{2} \) |
| 17 | \( 1 + 5.26T + 17T^{2} \) |
| 19 | \( 1 - 0.118T + 19T^{2} \) |
| 23 | \( 1 + 7.51T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 5.38T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 0.378T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 4.33T + 53T^{2} \) |
| 59 | \( 1 - 8.88T + 59T^{2} \) |
| 61 | \( 1 + 6.67T + 61T^{2} \) |
| 67 | \( 1 + 3.42T + 67T^{2} \) |
| 71 | \( 1 + 7.25T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 0.278T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 - 7.20T + 97T^{2} \) |
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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
−8.883702695039559059444371882999, −8.057542173334503137192009596773, −6.91404047372085253518818573031, −6.03489172324340848353693291651, −5.76871231197843749153784086057, −4.49952173671772398963229690481, −4.23977546866920968075511589140, −2.84616089290145668220169416807, −2.09702516545248598812546675583, −1.12135439095948135937018372453,
1.12135439095948135937018372453, 2.09702516545248598812546675583, 2.84616089290145668220169416807, 4.23977546866920968075511589140, 4.49952173671772398963229690481, 5.76871231197843749153784086057, 6.03489172324340848353693291651, 6.91404047372085253518818573031, 8.057542173334503137192009596773, 8.883702695039559059444371882999