L(s) = 1 | + 2.39i·2-s + (−0.796 + 1.53i)3-s − 3.73·4-s + (−2.17 + 0.517i)5-s + (−3.68 − 1.90i)6-s + 3.38·7-s − 4.14i·8-s + (−1.73 − 2.44i)9-s + (−1.23 − 5.20i)10-s + (2.69 + 1.93i)11-s + (2.97 − 5.74i)12-s − 3.38·13-s + 8.10i·14-s + (0.935 − 3.75i)15-s + 2.46·16-s + 1.75i·17-s + ⋯ |
L(s) = 1 | + 1.69i·2-s + (−0.459 + 0.888i)3-s − 1.86·4-s + (−0.972 + 0.231i)5-s + (−1.50 − 0.778i)6-s + 1.27·7-s − 1.46i·8-s + (−0.577 − 0.816i)9-s + (−0.391 − 1.64i)10-s + (0.812 + 0.582i)11-s + (0.857 − 1.65i)12-s − 0.939·13-s + 2.16i·14-s + (0.241 − 0.970i)15-s + 0.616·16-s + 0.425i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.264932 - 0.720212i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.264932 - 0.720212i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.796 - 1.53i)T \) |
| 5 | \( 1 + (2.17 - 0.517i)T \) |
| 11 | \( 1 + (-2.69 - 1.93i)T \) |
good | 2 | \( 1 - 2.39iT - 2T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 13 | \( 1 + 3.38T + 13T^{2} \) |
| 17 | \( 1 - 1.75iT - 17T^{2} \) |
| 19 | \( 1 - 3.81iT - 19T^{2} \) |
| 23 | \( 1 + 2.75T + 23T^{2} \) |
| 29 | \( 1 - 1.97T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 8.40iT - 37T^{2} \) |
| 41 | \( 1 - 7.36T + 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 47 | \( 1 + 1.59T + 47T^{2} \) |
| 53 | \( 1 - 8.70T + 53T^{2} \) |
| 59 | \( 1 - 4.89iT - 59T^{2} \) |
| 61 | \( 1 - 10.4iT - 61T^{2} \) |
| 67 | \( 1 + 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 3.58iT - 71T^{2} \) |
| 73 | \( 1 - 3.38T + 73T^{2} \) |
| 79 | \( 1 + 11.4iT - 79T^{2} \) |
| 83 | \( 1 + 10.0iT - 83T^{2} \) |
| 89 | \( 1 + 9.52iT - 89T^{2} \) |
| 97 | \( 1 + 8.40iT - 97T^{2} \) |
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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
−14.22897417568273297718378169247, −12.28204539066443468750742654764, −11.52465084585351378622048518272, −10.29105613974448721157639303909, −9.033343388931633552723749520903, −8.061574887270757704731427732041, −7.23917366348540719896707939036, −5.97905493101049597150615380380, −4.76066331916478024322866840173, −4.12966275283505634739467567166,
0.827835580779089949197718874259, 2.37769579825862749801327230172, 4.12027103152327470634479911981, 5.18706024857247283283564685353, 7.19125036833692301945282477814, 8.244902995361861249866736432946, 9.257562491442301571604137461036, 10.97267299146818363057053550942, 11.25711673683961541399011045110, 12.07966847303622077064938313391