| L(s) = 1 | + (22.6 + 39.1i)2-s + (−33.7 − 19.5i)3-s + (−1.02e3 + 1.77e3i)4-s + (2.48e3 − 1.43e3i)5-s − 1.76e3i·6-s + (−4.45e4 + 1.08e5i)7-s − 9.26e4·8-s + (−2.64e5 − 4.58e5i)9-s + (1.12e5 + 6.48e4i)10-s + (−1.70e6 + 2.95e6i)11-s + (6.91e4 − 3.99e4i)12-s − 6.97e6i·13-s + (−5.27e6 + 7.17e5i)14-s − 1.11e5·15-s + (−2.09e6 − 3.63e6i)16-s + (−2.47e7 − 1.42e7i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.0463 − 0.0267i)3-s + (−0.249 + 0.433i)4-s + (0.158 − 0.0917i)5-s − 0.0378i·6-s + (−0.378 + 0.925i)7-s − 0.353·8-s + (−0.498 − 0.863i)9-s + (0.112 + 0.0648i)10-s + (−0.963 + 1.66i)11-s + (0.0231 − 0.0133i)12-s − 1.44i·13-s + (−0.700 + 0.0952i)14-s − 0.00981·15-s + (−0.125 − 0.216i)16-s + (−1.02 − 0.591i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.0839913 - 0.637658i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0839913 - 0.637658i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-22.6 - 39.1i)T \) |
| 7 | \( 1 + (4.45e4 - 1.08e5i)T \) |
| good | 3 | \( 1 + (33.7 + 19.5i)T + (2.65e5 + 4.60e5i)T^{2} \) |
| 5 | \( 1 + (-2.48e3 + 1.43e3i)T + (1.22e8 - 2.11e8i)T^{2} \) |
| 11 | \( 1 + (1.70e6 - 2.95e6i)T + (-1.56e12 - 2.71e12i)T^{2} \) |
| 13 | \( 1 + 6.97e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (2.47e7 + 1.42e7i)T + (2.91e14 + 5.04e14i)T^{2} \) |
| 19 | \( 1 + (4.55e7 - 2.62e7i)T + (1.10e15 - 1.91e15i)T^{2} \) |
| 23 | \( 1 + (1.55e7 + 2.70e7i)T + (-1.09e16 + 1.89e16i)T^{2} \) |
| 29 | \( 1 - 4.56e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + (-1.51e9 - 8.76e8i)T + (3.93e17 + 6.82e17i)T^{2} \) |
| 37 | \( 1 + (-9.08e8 - 1.57e9i)T + (-3.29e18 + 5.70e18i)T^{2} \) |
| 41 | \( 1 - 5.14e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 7.27e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (-3.70e9 + 2.14e9i)T + (5.80e19 - 1.00e20i)T^{2} \) |
| 53 | \( 1 + (-3.61e9 + 6.25e9i)T + (-2.45e20 - 4.25e20i)T^{2} \) |
| 59 | \( 1 + (4.46e9 + 2.57e9i)T + (8.89e20 + 1.54e21i)T^{2} \) |
| 61 | \( 1 + (-4.42e9 + 2.55e9i)T + (1.32e21 - 2.29e21i)T^{2} \) |
| 67 | \( 1 + (-3.79e10 + 6.57e10i)T + (-4.09e21 - 7.08e21i)T^{2} \) |
| 71 | \( 1 + 1.13e11T + 1.64e22T^{2} \) |
| 73 | \( 1 + (-1.51e11 - 8.72e10i)T + (1.14e22 + 1.98e22i)T^{2} \) |
| 79 | \( 1 + (1.13e11 + 1.95e11i)T + (-2.95e22 + 5.11e22i)T^{2} \) |
| 83 | \( 1 - 4.13e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (2.25e11 - 1.30e11i)T + (1.23e23 - 2.13e23i)T^{2} \) |
| 97 | \( 1 + 4.10e11iT - 6.93e23T^{2} \) |
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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
−17.48606999316317701719716828940, −15.52994511467382337499970960188, −15.06707922421962435141331541418, −13.08740776276861287747876782597, −12.16371381291721174669223653413, −9.908885684575601012315800654763, −8.308181609298099676411742873653, −6.48172897018149596351097999522, −5.00988230373660620724522464444, −2.73635636688209454012705066053,
0.22908995015466285606503981543, 2.44934220383812357917504357975, 4.31890425845896380636999119439, 6.25893299892068418659202093966, 8.479247160650226456347244876111, 10.43525082063634029758715742317, 11.33140973447337487038466588999, 13.40535837829066377734905689146, 13.88999615024165490220530281873, 15.96814541723296374037732151238
