| L(s) = 1 | + (1.30 + 1.51i)2-s + (2.26 + 1.45i)3-s + (−0.569 + 3.95i)4-s + (−3.30 + 7.24i)5-s + (0.767 + 5.34i)6-s + (19.2 + 5.64i)7-s + (−6.73 + 4.32i)8-s + (−8.19 − 17.9i)9-s + (−15.2 + 4.48i)10-s + (−2.55 + 2.95i)11-s + (−7.06 + 8.15i)12-s + (26.0 − 7.63i)13-s + (16.6 + 36.4i)14-s + (−18.0 + 11.6i)15-s + (−15.3 − 4.50i)16-s + (−9.32 − 64.8i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (0.436 + 0.280i)3-s + (−0.0711 + 0.494i)4-s + (−0.295 + 0.647i)5-s + (0.0522 + 0.363i)6-s + (1.03 + 0.304i)7-s + (−0.297 + 0.191i)8-s + (−0.303 − 0.664i)9-s + (−0.483 + 0.141i)10-s + (−0.0701 + 0.0809i)11-s + (−0.170 + 0.196i)12-s + (0.554 − 0.162i)13-s + (0.317 + 0.695i)14-s + (−0.311 + 0.199i)15-s + (−0.239 − 0.0704i)16-s + (−0.133 − 0.925i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.50235 + 1.09130i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50235 + 1.09130i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 - 1.51i)T \) |
| 23 | \( 1 + (41.7 + 102. i)T \) |
| good | 3 | \( 1 + (-2.26 - 1.45i)T + (11.2 + 24.5i)T^{2} \) |
| 5 | \( 1 + (3.30 - 7.24i)T + (-81.8 - 94.4i)T^{2} \) |
| 7 | \( 1 + (-19.2 - 5.64i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (2.55 - 2.95i)T + (-189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-26.0 + 7.63i)T + (1.84e3 - 1.18e3i)T^{2} \) |
| 17 | \( 1 + (9.32 + 64.8i)T + (-4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-9.63 + 67.0i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-4.57 - 31.8i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (136. - 87.4i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (-87.7 - 192. i)T + (-3.31e4 + 3.82e4i)T^{2} \) |
| 41 | \( 1 + (181. - 397. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-109. - 70.4i)T + (3.30e4 + 7.23e4i)T^{2} \) |
| 47 | \( 1 + 201.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-218. - 64.0i)T + (1.25e5 + 8.04e4i)T^{2} \) |
| 59 | \( 1 + (716. - 210. i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-466. + 299. i)T + (9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (343. + 396. i)T + (-4.28e4 + 2.97e5i)T^{2} \) |
| 71 | \( 1 + (569. + 657. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-64.6 + 449. i)T + (-3.73e5 - 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-743. + 218. i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-155. - 339. i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-725. - 466. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (535. - 1.17e3i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
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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
−15.10626600793142865029699865029, −14.69330915337798649500151683487, −13.56748514283722113571287215184, −11.97357729687650909706312727858, −10.98036767926848119725768105130, −9.138624406543897920385712214864, −7.989941449986323987674354138254, −6.55938537932836826766313785581, −4.81518888314222033270441683458, −3.08894132121596549876563561891,
1.73631644389557103395856118050, 4.02190317362014137927703701108, 5.51645926120816796352792204801, 7.75837849944676141683173699471, 8.733681913609245082569827936234, 10.55754143761734228908002462839, 11.57272706771296705525500854799, 12.81893130986573307385539454422, 13.83723385956971909543767974221, 14.68478107787060891481700850738
