| L(s) = 1 | + 0.792i·2-s − i·3-s + 1.37·4-s + i·5-s + 0.792·6-s + (0.792 − 2.52i)7-s + 2.67i·8-s − 9-s − 0.792·10-s − 3.31i·11-s − 1.37i·12-s + (2 + 0.627i)14-s + 15-s + 0.627·16-s − 4.10·17-s − 0.792i·18-s + ⋯ |
| L(s) = 1 | + 0.560i·2-s − 0.577i·3-s + 0.686·4-s + 0.447i·5-s + 0.323·6-s + (0.299 − 0.954i)7-s + 0.944i·8-s − 0.333·9-s − 0.250·10-s − 1.00i·11-s − 0.396i·12-s + (0.534 + 0.167i)14-s + 0.258·15-s + 0.156·16-s − 0.996·17-s − 0.186i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.060467455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.060467455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-0.792 + 2.52i)T \) |
| 11 | \( 1 + 3.31iT \) |
| good | 2 | \( 1 - 0.792iT - 2T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.10T + 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 23 | \( 1 - 9.11T + 23T^{2} \) |
| 29 | \( 1 + 5.98iT - 29T^{2} \) |
| 31 | \( 1 + 6.74iT - 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 4.10iT - 43T^{2} \) |
| 47 | \( 1 - 12.7iT - 47T^{2} \) |
| 53 | \( 1 + 1.62T + 53T^{2} \) |
| 59 | \( 1 - 1.62iT - 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 4.74T + 67T^{2} \) |
| 71 | \( 1 - 3.25T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 0.294iT - 79T^{2} \) |
| 83 | \( 1 - 9.45T + 83T^{2} \) |
| 89 | \( 1 + 8.37iT - 89T^{2} \) |
| 97 | \( 1 - 8.37iT - 97T^{2} \) |
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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
−9.755186620785912957170255995797, −8.647597278523539819189960615524, −7.80173395207462702778086391798, −7.28005685644074498730982066831, −6.52880978086855045291680256298, −5.85670562921955447844611004215, −4.73430122471734740874027887560, −3.36631483709494395781839278043, −2.42588545536598922132493123691, −0.980622842476732076888537468571,
1.43916738927625094013094006236, 2.53624272790781583890438394803, 3.43060565841168766171213976870, 4.78603232007975838240666149176, 5.26723645806604620743622794291, 6.57445377712207720710312311665, 7.24950947822185493420011063663, 8.470629998743700181839715881760, 9.167361657761093750437056728649, 9.840923569381485126144729590646
