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
−9.840923569381485126144729590646, −9.167361657761093750437056728649, −8.470629998743700181839715881760, −7.24950947822185493420011063663, −6.57445377712207720710312311665, −5.26723645806604620743622794291, −4.78603232007975838240666149176, −3.43060565841168766171213976870, −2.53624272790781583890438394803, −1.43916738927625094013094006236,
0.980622842476732076888537468571, 2.42588545536598922132493123691, 3.36631483709494395781839278043, 4.73430122471734740874027887560, 5.85670562921955447844611004215, 6.52880978086855045291680256298, 7.28005685644074498730982066831, 7.80173395207462702778086391798, 8.647597278523539819189960615524, 9.755186620785912957170255995797