| L(s) = 1 | + (−0.694 + 3.93i)2-s + (−22.0 − 18.5i)3-s + (−15.0 − 5.47i)4-s + (4.74 − 1.72i)5-s + (88.3 − 74.1i)6-s + (104. + 181. i)7-s + (32 − 55.4i)8-s + (102. + 579. i)9-s + (3.50 + 19.8i)10-s + (208. − 361. i)11-s + (230. + 399. i)12-s + (19.4 − 16.2i)13-s + (−787. + 286. i)14-s + (−136. − 49.7i)15-s + (196. + 164. i)16-s + (−141. + 805. i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−1.41 − 1.18i)3-s + (−0.469 − 0.171i)4-s + (0.0848 − 0.0308i)5-s + (1.00 − 0.840i)6-s + (0.807 + 1.39i)7-s + (0.176 − 0.306i)8-s + (0.420 + 2.38i)9-s + (0.0110 + 0.0629i)10-s + (0.519 − 0.899i)11-s + (0.462 + 0.800i)12-s + (0.0318 − 0.0267i)13-s + (−1.07 + 0.390i)14-s + (−0.156 − 0.0571i)15-s + (0.191 + 0.160i)16-s + (−0.119 + 0.675i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.744061 + 0.471544i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.744061 + 0.471544i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.694 - 3.93i)T \) |
| 19 | \( 1 + (-916. - 1.27e3i)T \) |
| good | 3 | \( 1 + (22.0 + 18.5i)T + (42.1 + 239. i)T^{2} \) |
| 5 | \( 1 + (-4.74 + 1.72i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (-104. - 181. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-208. + 361. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-19.4 + 16.2i)T + (6.44e4 - 3.65e5i)T^{2} \) |
| 17 | \( 1 + (141. - 805. i)T + (-1.33e6 - 4.85e5i)T^{2} \) |
| 23 | \( 1 + (-3.20e3 - 1.16e3i)T + (4.93e6 + 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-881. - 4.99e3i)T + (-1.92e7 + 7.01e6i)T^{2} \) |
| 31 | \( 1 + (2.11e3 + 3.65e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 7.70e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-2.84e3 - 2.38e3i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + (-1.44e4 + 5.26e3i)T + (1.12e8 - 9.44e7i)T^{2} \) |
| 47 | \( 1 + (-2.65e3 - 1.50e4i)T + (-2.15e8 + 7.84e7i)T^{2} \) |
| 53 | \( 1 + (-2.00e4 - 7.28e3i)T + (3.20e8 + 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-2.17e3 + 1.23e4i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (3.10e4 + 1.13e4i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-9.79e3 - 5.55e4i)T + (-1.26e9 + 4.61e8i)T^{2} \) |
| 71 | \( 1 + (1.79e3 - 652. i)T + (1.38e9 - 1.15e9i)T^{2} \) |
| 73 | \( 1 + (-3.08e4 - 2.58e4i)T + (3.59e8 + 2.04e9i)T^{2} \) |
| 79 | \( 1 + (4.18e4 + 3.51e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (3.69e4 + 6.40e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-6.19e4 + 5.20e4i)T + (9.69e8 - 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-2.30e4 + 1.30e5i)T + (-8.06e9 - 2.93e9i)T^{2} \) |
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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
−15.73969122957105017581427126251, −14.31040228540639719385739204879, −12.95584174649137155882788384667, −11.91116526696259549891475918846, −11.01292679557860240653228990139, −8.799536398712345497708101405132, −7.49306802757154120195739102971, −6.00039442957319868651729857390, −5.37909025742814371655013057750, −1.41586612552271309830632325911,
0.72972913479443567273542932207, 4.11286072168723425926519920847, 4.97512734959640342360513752048, 7.03199474732414335251653191085, 9.390652653432054317924723067832, 10.41590783536632897432153672198, 11.17916145441380475029745676490, 12.12373261045564552376700515208, 13.85287334232734123891155495855, 15.23209519001045857106722193790
