| L(s) = 1 | + 0.942·2-s + 2.26·3-s − 1.11·4-s + 4.16·5-s + 2.13·6-s − 0.914·7-s − 2.93·8-s + 2.15·9-s + 3.92·10-s − 11-s − 2.52·12-s + 6.85·13-s − 0.861·14-s + 9.44·15-s − 0.537·16-s − 7.07·17-s + 2.02·18-s − 0.758·19-s − 4.63·20-s − 2.07·21-s − 0.942·22-s + 0.119·23-s − 6.65·24-s + 12.3·25-s + 6.45·26-s − 1.92·27-s + 1.01·28-s + ⋯ |
| L(s) = 1 | + 0.666·2-s + 1.31·3-s − 0.556·4-s + 1.86·5-s + 0.872·6-s − 0.345·7-s − 1.03·8-s + 0.716·9-s + 1.23·10-s − 0.301·11-s − 0.728·12-s + 1.90·13-s − 0.230·14-s + 2.43·15-s − 0.134·16-s − 1.71·17-s + 0.477·18-s − 0.173·19-s − 1.03·20-s − 0.452·21-s − 0.200·22-s + 0.0249·23-s − 1.35·24-s + 2.46·25-s + 1.26·26-s − 0.371·27-s + 0.192·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.322752885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.322752885\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.942T + 2T^{2} \) |
| 3 | \( 1 - 2.26T + 3T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 7 | \( 1 + 0.914T + 7T^{2} \) |
| 13 | \( 1 - 6.85T + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 + 0.758T + 19T^{2} \) |
| 23 | \( 1 - 0.119T + 23T^{2} \) |
| 29 | \( 1 + 6.47T + 29T^{2} \) |
| 31 | \( 1 - 8.12T + 31T^{2} \) |
| 37 | \( 1 + 3.59T + 37T^{2} \) |
| 41 | \( 1 + 3.48T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 3.93T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 + 8.02T + 59T^{2} \) |
| 67 | \( 1 + 9.13T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 6.32T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 0.929T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−10.22951871245559458502932268409, −9.430388269970113985055980005149, −8.830810404222357544462606331256, −8.378929631388163702611603320201, −6.57584259608000493675182449828, −6.05467040360897302418758953585, −4.98984391573624735480689383503, −3.77817977908686544874884306682, −2.84909950770955815975249906409, −1.80831441279549431176785125938,
1.80831441279549431176785125938, 2.84909950770955815975249906409, 3.77817977908686544874884306682, 4.98984391573624735480689383503, 6.05467040360897302418758953585, 6.57584259608000493675182449828, 8.378929631388163702611603320201, 8.830810404222357544462606331256, 9.430388269970113985055980005149, 10.22951871245559458502932268409
