| L(s) = 1 | + (2.93 + 0.442i)3-s + (−0.492 − 0.335i)5-s + (−2.18 + 3.77i)7-s + (5.54 + 1.71i)9-s + (0.452 + 1.98i)11-s + (−0.130 + 1.73i)13-s + (−1.29 − 1.20i)15-s + (1.21 − 0.826i)17-s + (2.07 − 0.641i)19-s + (−8.07 + 10.1i)21-s + (−1.99 + 1.85i)23-s + (−1.69 − 4.32i)25-s + (7.48 + 3.60i)27-s + (7.48 − 1.12i)29-s + (0.208 − 0.530i)31-s + ⋯ |
| L(s) = 1 | + (1.69 + 0.255i)3-s + (−0.220 − 0.150i)5-s + (−0.824 + 1.42i)7-s + (1.84 + 0.570i)9-s + (0.136 + 0.598i)11-s + (−0.0361 + 0.482i)13-s + (−0.334 − 0.310i)15-s + (0.294 − 0.200i)17-s + (0.476 − 0.147i)19-s + (−1.76 + 2.20i)21-s + (−0.416 + 0.386i)23-s + (−0.339 − 0.864i)25-s + (1.44 + 0.693i)27-s + (1.39 − 0.209i)29-s + (0.0373 − 0.0952i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.08199 + 1.13959i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08199 + 1.13959i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (3.81 + 5.32i)T \) |
| good | 3 | \( 1 + (-2.93 - 0.442i)T + (2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (0.492 + 0.335i)T + (1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (2.18 - 3.77i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.452 - 1.98i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (0.130 - 1.73i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-1.21 + 0.826i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-2.07 + 0.641i)T + (15.6 - 10.7i)T^{2} \) |
| 23 | \( 1 + (1.99 - 1.85i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-7.48 + 1.12i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (-0.208 + 0.530i)T + (-22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (-1.98 - 3.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.32 - 1.66i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-2.08 + 9.12i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (0.441 + 5.88i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-4.60 - 2.21i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (3.57 + 9.10i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-0.506 + 0.156i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (3.91 + 3.63i)T + (5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (1.05 - 14.0i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-1.82 + 3.16i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (10.2 + 1.54i)T + (79.3 + 24.4i)T^{2} \) |
| 89 | \( 1 + (-5.97 - 0.899i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + (2.28 + 9.99i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−9.931814786675345423308126852828, −9.781759849356512681297664524992, −8.770015706312616239149597408424, −8.365762956497035674406807086710, −7.30324715654056237875703200600, −6.28733852160979229939443031191, −4.95361730512184635722321403857, −3.82811900251160219676151988406, −2.86726721308361773948244870622, −2.07269456061546044092897341494,
1.13951443013461735967755102305, 2.87822381407819067562000395074, 3.48532147652383221611553019608, 4.33863913734002823124231098938, 6.13167151630254676506612054255, 7.18779100005085564268120321741, 7.70537533320006039776535577122, 8.512071924652304124569928198890, 9.470054947964146226919980529093, 10.12171739669534064265231120389
