| L(s) = 1 | + (1.17 − 1.61i)2-s + (−5.96 + 1.93i)3-s + (−1.23 − 3.80i)4-s + (−3.87 + 11.9i)6-s + 9.80i·7-s + (−7.60 − 2.47i)8-s + (10.0 − 7.28i)9-s + (23.6 + 17.2i)11-s + (14.7 + 20.3i)12-s + (−36.2 − 49.8i)13-s + (15.8 + 11.5i)14-s + (−12.9 + 9.40i)16-s + (87.6 + 28.4i)17-s − 24.7i·18-s + (4.22 − 13.0i)19-s + ⋯ |
| L(s) = 1 | + (0.415 − 0.572i)2-s + (−1.14 + 0.373i)3-s + (−0.154 − 0.475i)4-s + (−0.263 + 0.812i)6-s + 0.529i·7-s + (−0.336 − 0.109i)8-s + (0.371 − 0.269i)9-s + (0.649 + 0.471i)11-s + (0.354 + 0.488i)12-s + (−0.772 − 1.06i)13-s + (0.303 + 0.220i)14-s + (−0.202 + 0.146i)16-s + (1.24 + 0.406i)17-s − 0.324i·18-s + (0.0510 − 0.157i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.05582 - 0.747700i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05582 - 0.747700i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.17 + 1.61i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (5.96 - 1.93i)T + (21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 - 9.80iT - 343T^{2} \) |
| 11 | \( 1 + (-23.6 - 17.2i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (36.2 + 49.8i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-87.6 - 28.4i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-4.22 + 13.0i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-91.7 + 126. i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (7.04 + 21.6i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-55.2 + 169. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (246. + 339. i)T + (-1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-27.3 + 19.8i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 99.2iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-575. + 187. i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-85.2 + 27.7i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (192. - 140. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-484. - 352. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-788. - 256. i)T + (2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (76.5 + 235. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (604. - 831. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (136. + 421. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (993. + 322. i)T + (4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (441. + 320. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (874. - 284. i)T + (7.38e5 - 5.36e5i)T^{2} \) |
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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
−11.54955502241320180322789701412, −10.52434752233645768438859644771, −9.973036815143249292743489997953, −8.730768447918203414157894327529, −7.23637890979833360580871187895, −5.85542272629220531394846988945, −5.30076908327283836982917147914, −4.13507961653624780407576502883, −2.57337752382516151456411847100, −0.65160725792090897833458497742,
1.10318289672045128680687406337, 3.42735096486434729786643503871, 4.83395239325844519237676861470, 5.70382861495541676006973396090, 6.79919935855162335297764560038, 7.33855441417168391563694931284, 8.809317387621561475738385036887, 9.944629495343475835782706841472, 11.19373406928193844040403303533, 11.91635040841586656701275410035
