| L(s) = 1 | − 2.18i·2-s − 2.79·4-s + (2.18 − 0.456i)5-s − i·7-s + 1.73i·8-s + (−0.999 − 4.79i)10-s + 1.73·11-s − 4.79i·13-s − 2.18·14-s − 1.79·16-s + 5.65i·17-s + 6.79·19-s + (−6.10 + 1.27i)20-s − 3.79i·22-s − 4.83i·23-s + ⋯ |
| L(s) = 1 | − 1.54i·2-s − 1.39·4-s + (0.978 − 0.204i)5-s − 0.377i·7-s + 0.612i·8-s + (−0.316 − 1.51i)10-s + 0.522·11-s − 1.32i·13-s − 0.585·14-s − 0.447·16-s + 1.37i·17-s + 1.55·19-s + (−1.36 + 0.285i)20-s − 0.808i·22-s − 1.00i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.183952 - 1.78173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.183952 - 1.78173i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.18 + 0.456i)T \) |
| 7 | \( 1 + iT \) |
| good | 2 | \( 1 + 2.18iT - 2T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 4.79iT - 13T^{2} \) |
| 17 | \( 1 - 5.65iT - 17T^{2} \) |
| 19 | \( 1 - 6.79T + 19T^{2} \) |
| 23 | \( 1 + 4.83iT - 23T^{2} \) |
| 29 | \( 1 - 3.00T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 + 10.1iT - 37T^{2} \) |
| 41 | \( 1 + 9.11T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 1.73iT - 47T^{2} \) |
| 53 | \( 1 - 6.56iT - 53T^{2} \) |
| 59 | \( 1 + 7.74T + 59T^{2} \) |
| 61 | \( 1 - 4.79T + 61T^{2} \) |
| 67 | \( 1 - 1.37iT - 67T^{2} \) |
| 71 | \( 1 - 1.17T + 71T^{2} \) |
| 73 | \( 1 - 4.58iT - 73T^{2} \) |
| 79 | \( 1 + 3.37T + 79T^{2} \) |
| 83 | \( 1 - 17.4iT - 83T^{2} \) |
| 89 | \( 1 - 8.66T + 89T^{2} \) |
| 97 | \( 1 + 1.37iT - 97T^{2} \) |
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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
−9.928503009243101358688285162769, −9.181669731725352567909307609091, −8.355598914310530176143858037803, −7.12475426757440129105613010143, −5.96823054959308293016664039607, −5.10347683340262460186848219220, −3.90485548420766991901229049357, −3.06679691119750474647143765210, −1.90916921526486606349985009881, −0.902267585546710843733223853518,
1.74186733076079139804656394510, 3.24131273684592580492800073099, 4.84103183702620929939732749906, 5.34671325681116239254818031478, 6.32622768536883538669897581529, 6.93598913264344156759503374816, 7.60545138637303909155623167090, 8.888041750523860595397420983239, 9.285299980212244905657695405305, 9.946210416904947886659943418204
