L(s) = 1 | + (−0.334 + 0.192i)2-s + (−1.42 + 0.983i)3-s + (−0.925 + 1.60i)4-s + (−0.5 − 0.866i)5-s + (0.286 − 0.603i)6-s − 1.48i·8-s + (1.06 − 2.80i)9-s + (0.334 + 0.192i)10-s + (−2.20 − 1.27i)11-s + (−0.257 − 3.19i)12-s + 3.06i·13-s + (1.56 + 0.742i)15-s + (−1.56 − 2.71i)16-s + (3.23 − 5.59i)17-s + (0.185 + 1.14i)18-s + (1.03 − 0.597i)19-s + ⋯ |
L(s) = 1 | + (−0.236 + 0.136i)2-s + (−0.823 + 0.567i)3-s + (−0.462 + 0.801i)4-s + (−0.223 − 0.387i)5-s + (0.116 − 0.246i)6-s − 0.525i·8-s + (0.354 − 0.934i)9-s + (0.105 + 0.0609i)10-s + (−0.663 − 0.383i)11-s + (−0.0743 − 0.922i)12-s + 0.850i·13-s + (0.404 + 0.191i)15-s + (−0.391 − 0.677i)16-s + (0.783 − 1.35i)17-s + (0.0436 + 0.269i)18-s + (0.237 − 0.137i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.698359 + 0.00594313i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.698359 + 0.00594313i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.42 - 0.983i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.334 - 0.192i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (2.20 + 1.27i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.06iT - 13T^{2} \) |
| 17 | \( 1 + (-3.23 + 5.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.03 + 0.597i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.64 - 1.52i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.77iT - 29T^{2} \) |
| 31 | \( 1 + (-5.95 - 3.43i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.77 - 3.07i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.31T + 41T^{2} \) |
| 43 | \( 1 - 5.46T + 43T^{2} \) |
| 47 | \( 1 + (-1.61 - 2.78i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.4 - 6.62i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.98 + 3.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.08 + 4.67i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.75 + 3.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.921iT - 71T^{2} \) |
| 73 | \( 1 + (0.256 + 0.148i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.14 + 7.18i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.11T + 83T^{2} \) |
| 89 | \( 1 + (9.41 + 16.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.3iT - 97T^{2} \) |
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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
−10.15442922914283450321434056870, −9.584427015039543496317398860806, −8.755093697977272014115572686082, −7.81453976004449596597771441435, −7.00087739897362849570677226912, −5.81486803316535040968619310293, −4.83366187115884339303017699969, −4.12242898325568666001498972385, −2.96635230611218035642537637121, −0.59133220211617808048021546116,
0.975745446884842234141811430867, 2.34375578836881206917872302579, 4.04170509141804217095873228526, 5.30331360608488452787096068467, 5.77015159943415751327530106767, 6.80869727924567629764084991661, 7.84247405294167920090046577387, 8.488400138375706490860677207975, 9.995437840079344869923357959520, 10.34230074742522565153896960070