| L(s) = 1 | + 2.31·2-s − 3-s + 3.34·4-s + 1.46·5-s − 2.31·6-s + 7-s + 3.10·8-s + 9-s + 3.37·10-s + 0.0565·11-s − 3.34·12-s + 2.31·14-s − 1.46·15-s + 0.489·16-s + 5.75·17-s + 2.31·18-s − 0.807·19-s + 4.88·20-s − 21-s + 0.130·22-s + 6.01·23-s − 3.10·24-s − 2.86·25-s − 27-s + 3.34·28-s + 3.22·29-s − 3.37·30-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 0.577·3-s + 1.67·4-s + 0.653·5-s − 0.943·6-s + 0.377·7-s + 1.09·8-s + 0.333·9-s + 1.06·10-s + 0.0170·11-s − 0.965·12-s + 0.617·14-s − 0.377·15-s + 0.122·16-s + 1.39·17-s + 0.544·18-s − 0.185·19-s + 1.09·20-s − 0.218·21-s + 0.0278·22-s + 1.25·23-s − 0.633·24-s − 0.573·25-s − 0.192·27-s + 0.631·28-s + 0.598·29-s − 0.616·30-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
| \(L(1)\) |
\(\approx\) |
\(5.142499701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.142499701\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.31T + 2T^{2} \) |
| 5 | \( 1 - 1.46T + 5T^{2} \) |
| 11 | \( 1 - 0.0565T + 11T^{2} \) |
| 17 | \( 1 - 5.75T + 17T^{2} \) |
| 19 | \( 1 + 0.807T + 19T^{2} \) |
| 23 | \( 1 - 6.01T + 23T^{2} \) |
| 29 | \( 1 - 3.22T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 0.780T + 37T^{2} \) |
| 41 | \( 1 + 5.30T + 41T^{2} \) |
| 43 | \( 1 - 2.38T + 43T^{2} \) |
| 47 | \( 1 + 3.32T + 47T^{2} \) |
| 53 | \( 1 + 9.52T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 8.88T + 61T^{2} \) |
| 67 | \( 1 - 7.80T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 7.70T + 83T^{2} \) |
| 89 | \( 1 - 8.12T + 89T^{2} \) |
| 97 | \( 1 - 3.93T + 97T^{2} \) |
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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
−8.403938700822821636256854058887, −7.52690115797734966106375253040, −6.63073000854529905358492698500, −6.17859883492543660810161122791, −5.20635794995178709291517769641, −5.10758672005025746232921272062, −4.04858009674293373722490210945, −3.21294912213694689869037096451, −2.29588308444727451007650046405, −1.16992978316961225854337825540,
1.16992978316961225854337825540, 2.29588308444727451007650046405, 3.21294912213694689869037096451, 4.04858009674293373722490210945, 5.10758672005025746232921272062, 5.20635794995178709291517769641, 6.17859883492543660810161122791, 6.63073000854529905358492698500, 7.52690115797734966106375253040, 8.403938700822821636256854058887
