| L(s) = 1 | + (3.75 − 1.36i)2-s + (−0.383 + 2.17i)3-s + (12.2 − 10.2i)4-s + (53.6 + 44.9i)5-s + (1.53 + 8.69i)6-s + (−9.54 − 16.5i)7-s + (32.0 − 55.4i)8-s + (223. + 81.4i)9-s + (263. + 95.7i)10-s + (221. − 383. i)11-s + (17.6 + 30.5i)12-s + (66.2 + 375. i)13-s + (−58.4 − 49.0i)14-s + (−118. + 99.2i)15-s + (44.4 − 252. i)16-s + (−1.38e3 + 504. i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.0245 + 0.139i)3-s + (0.383 − 0.321i)4-s + (0.959 + 0.804i)5-s + (0.0173 + 0.0985i)6-s + (−0.0736 − 0.127i)7-s + (0.176 − 0.306i)8-s + (0.920 + 0.335i)9-s + (0.831 + 0.302i)10-s + (0.552 − 0.956i)11-s + (0.0353 + 0.0613i)12-s + (0.108 + 0.616i)13-s + (−0.0797 − 0.0669i)14-s + (−0.135 + 0.113i)15-s + (0.0434 − 0.246i)16-s + (−1.16 + 0.423i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0320i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.63647 + 0.0422775i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.63647 + 0.0422775i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.75 + 1.36i)T \) |
| 19 | \( 1 + (-548. + 1.47e3i)T \) |
| good | 3 | \( 1 + (0.383 - 2.17i)T + (-228. - 83.1i)T^{2} \) |
| 5 | \( 1 + (-53.6 - 44.9i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (9.54 + 16.5i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-221. + 383. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-66.2 - 375. i)T + (-3.48e5 + 1.26e5i)T^{2} \) |
| 17 | \( 1 + (1.38e3 - 504. i)T + (1.08e6 - 9.12e5i)T^{2} \) |
| 23 | \( 1 + (2.89e3 - 2.42e3i)T + (1.11e6 - 6.33e6i)T^{2} \) |
| 29 | \( 1 + (5.18e3 + 1.88e3i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (1.99e3 + 3.45e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 6.45e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-219. + 1.24e3i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + (-2.99e3 - 2.51e3i)T + (2.55e7 + 1.44e8i)T^{2} \) |
| 47 | \( 1 + (-1.62e4 - 5.91e3i)T + (1.75e8 + 1.47e8i)T^{2} \) |
| 53 | \( 1 + (-1.46e4 + 1.23e4i)T + (7.26e7 - 4.11e8i)T^{2} \) |
| 59 | \( 1 + (-2.73e4 + 9.93e3i)T + (5.47e8 - 4.59e8i)T^{2} \) |
| 61 | \( 1 + (2.65e4 - 2.23e4i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (2.32e4 + 8.45e3i)T + (1.03e9 + 8.67e8i)T^{2} \) |
| 71 | \( 1 + (4.46e4 + 3.74e4i)T + (3.13e8 + 1.77e9i)T^{2} \) |
| 73 | \( 1 + (5.47e3 - 3.10e4i)T + (-1.94e9 - 7.09e8i)T^{2} \) |
| 79 | \( 1 + (1.18e3 - 6.70e3i)T + (-2.89e9 - 1.05e9i)T^{2} \) |
| 83 | \( 1 + (3.03e4 + 5.24e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-2.00e4 - 1.13e5i)T + (-5.24e9 + 1.90e9i)T^{2} \) |
| 97 | \( 1 + (9.07e4 - 3.30e4i)T + (6.57e9 - 5.51e9i)T^{2} \) |
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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
−15.12044293104898394836101902537, −13.79165911286585784290825775557, −13.35649978900455698867199733485, −11.53364317000580677448903488017, −10.53628483744934641214964264968, −9.317758830209723908981656361303, −7.03459069475169103070961335671, −5.86966880261792672029258453973, −3.96443188328375491489922624207, −2.02641849901378694533553545013,
1.78092204518810082035380332489, 4.29494366635097203743201414296, 5.76257786823303696088099311895, 7.15908948852356729200681104781, 9.011435461432290567468291009386, 10.22996681650346025910469956210, 12.20950424744660456068071217121, 12.86600237908196027819616176189, 13.96178033805270122757260516448, 15.24226401339165167282717587576
