| L(s) = 1 | + 2.69·2-s + 3-s + 5.28·4-s + 2.72·5-s + 2.69·6-s + 2.49·7-s + 8.87·8-s + 9-s + 7.35·10-s − 3.62·11-s + 5.28·12-s − 0.474·13-s + 6.72·14-s + 2.72·15-s + 13.3·16-s − 17-s + 2.69·18-s − 4.94·19-s + 14.4·20-s + 2.49·21-s − 9.78·22-s − 3.75·23-s + 8.87·24-s + 2.41·25-s − 1.28·26-s + 27-s + 13.1·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 0.577·3-s + 2.64·4-s + 1.21·5-s + 1.10·6-s + 0.941·7-s + 3.13·8-s + 0.333·9-s + 2.32·10-s − 1.09·11-s + 1.52·12-s − 0.131·13-s + 1.79·14-s + 0.703·15-s + 3.34·16-s − 0.242·17-s + 0.636·18-s − 1.13·19-s + 3.22·20-s + 0.543·21-s − 2.08·22-s − 0.782·23-s + 1.81·24-s + 0.483·25-s − 0.251·26-s + 0.192·27-s + 2.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.39323686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.39323686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 - 2.72T + 5T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 + 3.62T + 11T^{2} \) |
| 13 | \( 1 + 0.474T + 13T^{2} \) |
| 19 | \( 1 + 4.94T + 19T^{2} \) |
| 23 | \( 1 + 3.75T + 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 + 0.932T + 31T^{2} \) |
| 37 | \( 1 + 7.91T + 37T^{2} \) |
| 41 | \( 1 - 1.34T + 41T^{2} \) |
| 43 | \( 1 - 5.31T + 43T^{2} \) |
| 47 | \( 1 - 0.0922T + 47T^{2} \) |
| 53 | \( 1 - 4.00T + 53T^{2} \) |
| 59 | \( 1 + 5.89T + 59T^{2} \) |
| 61 | \( 1 + 1.66T + 61T^{2} \) |
| 67 | \( 1 - 4.37T + 67T^{2} \) |
| 71 | \( 1 + 1.72T + 71T^{2} \) |
| 73 | \( 1 + 1.59T + 73T^{2} \) |
| 83 | \( 1 - 5.23T + 83T^{2} \) |
| 89 | \( 1 - 4.91T + 89T^{2} \) |
| 97 | \( 1 - 8.22T + 97T^{2} \) |
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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
−8.152073129041633005451208898153, −7.54411606510088573743733068285, −6.71848190471620897038066458212, −5.90031298142200143450251042197, −5.40690082355854648601120052189, −4.70619120966798342606091717738, −3.99691081341550911141917329922, −2.98322917725217697467706524765, −2.00880775076008665147998392271, −1.93960952665039009227250759301,
1.93960952665039009227250759301, 2.00880775076008665147998392271, 2.98322917725217697467706524765, 3.99691081341550911141917329922, 4.70619120966798342606091717738, 5.40690082355854648601120052189, 5.90031298142200143450251042197, 6.71848190471620897038066458212, 7.54411606510088573743733068285, 8.152073129041633005451208898153
