| L(s) = 1 | − 1.53·2-s + 3.20·3-s + 0.365·4-s − 4.92·6-s + 2.65·7-s + 2.51·8-s + 7.26·9-s + 1.92·11-s + 1.16·12-s + 3.08·13-s − 4.07·14-s − 4.59·16-s + 7.25·17-s − 11.1·18-s − 0.894·19-s + 8.49·21-s − 2.96·22-s − 8.28·23-s + 8.05·24-s − 4.74·26-s + 13.6·27-s + 0.967·28-s − 1.36·29-s − 0.423·31-s + 2.04·32-s + 6.16·33-s − 11.1·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 1.84·3-s + 0.182·4-s − 2.01·6-s + 1.00·7-s + 0.888·8-s + 2.42·9-s + 0.580·11-s + 0.337·12-s + 0.856·13-s − 1.08·14-s − 1.14·16-s + 1.75·17-s − 2.63·18-s − 0.205·19-s + 1.85·21-s − 0.631·22-s − 1.72·23-s + 1.64·24-s − 0.931·26-s + 2.62·27-s + 0.182·28-s − 0.253·29-s − 0.0760·31-s + 0.360·32-s + 1.07·33-s − 1.91·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.221050991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.221050991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 3 | \( 1 - 3.20T + 3T^{2} \) |
| 7 | \( 1 - 2.65T + 7T^{2} \) |
| 11 | \( 1 - 1.92T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 - 7.25T + 17T^{2} \) |
| 19 | \( 1 + 0.894T + 19T^{2} \) |
| 23 | \( 1 + 8.28T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 + 0.423T + 31T^{2} \) |
| 37 | \( 1 + 9.26T + 37T^{2} \) |
| 41 | \( 1 + 2.33T + 41T^{2} \) |
| 43 | \( 1 + 7.88T + 43T^{2} \) |
| 47 | \( 1 + 7.99T + 47T^{2} \) |
| 53 | \( 1 + 2.67T + 53T^{2} \) |
| 59 | \( 1 - 9.32T + 59T^{2} \) |
| 67 | \( 1 + 0.784T + 67T^{2} \) |
| 71 | \( 1 + 0.582T + 71T^{2} \) |
| 73 | \( 1 - 4.43T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 3.71T + 83T^{2} \) |
| 89 | \( 1 + 2.10T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−9.346682371788577261763214731233, −8.404677613681831950571971122075, −8.232260636762759378230011864979, −7.65609199211563838596414350716, −6.68567661044892934406996196170, −5.16727481920965372199650552846, −4.02999345154857898477720056572, −3.43170125174099672460893025786, −1.89676473279013804143373427715, −1.40259706341567511641048595925,
1.40259706341567511641048595925, 1.89676473279013804143373427715, 3.43170125174099672460893025786, 4.02999345154857898477720056572, 5.16727481920965372199650552846, 6.68567661044892934406996196170, 7.65609199211563838596414350716, 8.232260636762759378230011864979, 8.404677613681831950571971122075, 9.346682371788577261763214731233
