L(s) = 1 | + (−0.685 − 5.61i)2-s + 28.9·3-s + (−31.0 + 7.69i)4-s + (−40.5 − 38.5i)5-s + (−19.8 − 162. i)6-s − 128. i·7-s + (64.5 + 169. i)8-s + 592.·9-s + (−188. + 253. i)10-s − 433. i·11-s + (−897. + 222. i)12-s + 78.5·13-s + (−719. + 87.8i)14-s + (−1.17e3 − 1.11e3i)15-s + (905. − 478. i)16-s + 1.48e3i·17-s + ⋯ |
L(s) = 1 | + (−0.121 − 0.992i)2-s + 1.85·3-s + (−0.970 + 0.240i)4-s + (−0.724 − 0.689i)5-s + (−0.224 − 1.84i)6-s − 0.988i·7-s + (0.356 + 0.934i)8-s + 2.43·9-s + (−0.596 + 0.802i)10-s − 1.08i·11-s + (−1.79 + 0.446i)12-s + 0.128·13-s + (−0.980 + 0.119i)14-s + (−1.34 − 1.27i)15-s + (0.884 − 0.466i)16-s + 1.24i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.21381 - 1.82297i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21381 - 1.82297i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.685 + 5.61i)T \) |
| 5 | \( 1 + (40.5 + 38.5i)T \) |
good | 3 | \( 1 - 28.9T + 243T^{2} \) |
| 7 | \( 1 + 128. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 433. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 78.5T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.48e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 98.3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.36e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 1.02e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.30e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.64e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.44e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.06e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.55e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.67e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.29e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.92e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.06e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.72e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 5.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.32e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.50e4iT - 8.58e9T^{2} \) |
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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
−14.36830217566274128997337025222, −13.50935160477743369128356061179, −12.70991451050142562678566329911, −10.99221680217266260695940490752, −9.628759956477008538144128086411, −8.477586353087711642113099706960, −7.81833131841994496218190895878, −4.19514939469109799682650970221, −3.32302135982018593520332729521, −1.26272580867447112173123678550,
2.69869577149243655531011324308, 4.39687187618698848402252934508, 6.91305126090579118578533852994, 7.894334671937143909094986177872, 8.923207968867038035750980182551, 9.951703058611029060696827619691, 12.33440854237290633751636120299, 13.68766489158543850441111611127, 14.71074782833192643874307766297, 15.22943415514232198121193338855