| L(s) = 1 | + i·2-s − 4-s + 1.73i·5-s − 4.73i·7-s − i·8-s − 1.73·10-s + 4.73i·11-s + 4.73·14-s + 16-s − 5.19·17-s + 1.26i·19-s − 1.73i·20-s − 4.73·22-s + 2.19·23-s + 2.00·25-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.774i·5-s − 1.78i·7-s − 0.353i·8-s − 0.547·10-s + 1.42i·11-s + 1.26·14-s + 0.250·16-s − 1.26·17-s + 0.290i·19-s − 0.387i·20-s − 1.00·22-s + 0.457·23-s + 0.400·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9261606618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9261606618\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 + 4.73iT - 7T^{2} \) |
| 11 | \( 1 - 4.73iT - 11T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 1.26iT - 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 2.53iT - 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 0.464iT - 41T^{2} \) |
| 43 | \( 1 + 6.19T + 43T^{2} \) |
| 47 | \( 1 + 1.26iT - 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 13.8iT - 59T^{2} \) |
| 61 | \( 1 - 4.80T + 61T^{2} \) |
| 67 | \( 1 - 10.7iT - 67T^{2} \) |
| 71 | \( 1 + 8.19iT - 71T^{2} \) |
| 73 | \( 1 - 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 2.53iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.990815706107538061250050632790, −8.081654031349349833089996492157, −7.18958337044036658506612967423, −7.00841506986143513427241358296, −6.43710574658513813850490025565, −5.10717238815327686419250091804, −4.38386767823403846317582874454, −3.82078439526043869015774741772, −2.61812365475114739270998892494, −1.26387557611753727194401765180,
0.30300188091268477826440440497, 1.66520142302483777679319300576, 2.64823604294301214148879193302, 3.31088023549051013013276678955, 4.60839319173876106184715581880, 5.15967365969020026069904649934, 5.94968681617951073683584634279, 6.63320076716790490822361960942, 8.151216046749202505879480909086, 8.580080998641710208433201245408
