| L(s) = 1 | + 0.472i·3-s − 4.35i·5-s − 7-s + 2.77·9-s + 5.41i·11-s + i·13-s + 2.05·15-s − 6.30·17-s − 2.90i·19-s − 0.472i·21-s − 1.16·23-s − 13.9·25-s + 2.72i·27-s − 0.400i·29-s − 8.74·31-s + ⋯ |
| L(s) = 1 | + 0.272i·3-s − 1.94i·5-s − 0.377·7-s + 0.925·9-s + 1.63i·11-s + 0.277i·13-s + 0.530·15-s − 1.52·17-s − 0.667i·19-s − 0.103i·21-s − 0.243·23-s − 2.78·25-s + 0.524i·27-s − 0.0743i·29-s − 1.57·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.002203826989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002203826989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - iT \) |
| good | 3 | \( 1 - 0.472iT - 3T^{2} \) |
| 5 | \( 1 + 4.35iT - 5T^{2} \) |
| 11 | \( 1 - 5.41iT - 11T^{2} \) |
| 17 | \( 1 + 6.30T + 17T^{2} \) |
| 19 | \( 1 + 2.90iT - 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 + 0.400iT - 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 + 0.159iT - 37T^{2} \) |
| 41 | \( 1 + 0.297T + 41T^{2} \) |
| 43 | \( 1 + 3.74iT - 43T^{2} \) |
| 47 | \( 1 - 3.08T + 47T^{2} \) |
| 53 | \( 1 + 2.19iT - 53T^{2} \) |
| 59 | \( 1 + 3.99iT - 59T^{2} \) |
| 61 | \( 1 - 3.08iT - 61T^{2} \) |
| 67 | \( 1 - 13.4iT - 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 2.84iT - 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 4.12T + 97T^{2} \) |
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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
−8.719103927469400288205062595872, −7.39313597823845682780885172931, −7.07102285869187107092093735045, −5.86102958066222630259729570763, −4.93180287015349469717003747823, −4.43451556914871068669078678486, −3.98804000837667942780285880408, −2.18174841863803327477496877901, −1.47843203669419098124179076763, −0.00065503696009288793228497953,
1.79804135033068477839675878481, 2.81332584298963757463106720827, 3.47728787768572387861921673264, 4.24276730086943180814439758862, 5.82474590793397048600924061379, 6.17957672312248829394085505253, 6.99195601426640412387788815115, 7.44021449927532061962706821188, 8.340609816399705072243888459236, 9.246891867358132007103951608448
