| L(s) = 1 | + (−1.23 − 3.80i)2-s + (−23.0 + 16.7i)3-s + (−12.9 + 9.40i)4-s + (−50.3 + 24.2i)5-s + (92.2 + 67.0i)6-s + 20.3·7-s + (51.7 + 37.6i)8-s + (175. − 541. i)9-s + (154. + 161. i)10-s + (−57.1 − 175. i)11-s + (140. − 433. i)12-s + (−42.6 + 131. i)13-s + (−25.2 − 77.5i)14-s + (754. − 1.40e3i)15-s + (79.1 − 243. i)16-s + (1.75e3 + 1.27e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−1.47 + 1.07i)3-s + (−0.404 + 0.293i)4-s + (−0.900 + 0.433i)5-s + (1.04 + 0.759i)6-s + 0.157·7-s + (0.286 + 0.207i)8-s + (0.723 − 2.22i)9-s + (0.488 + 0.511i)10-s + (−0.142 − 0.437i)11-s + (0.282 − 0.869i)12-s + (−0.0700 + 0.215i)13-s + (−0.0343 − 0.105i)14-s + (0.866 − 1.60i)15-s + (0.0772 − 0.237i)16-s + (1.47 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.355027 - 0.253048i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.355027 - 0.253048i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 5 | \( 1 + (50.3 - 24.2i)T \) |
| good | 3 | \( 1 + (23.0 - 16.7i)T + (75.0 - 231. i)T^{2} \) |
| 7 | \( 1 - 20.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + (57.1 + 175. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (42.6 - 131. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-1.75e3 - 1.27e3i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (1.88e3 + 1.36e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (627. + 1.93e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-1.24e3 + 903. i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-5.98e3 - 4.34e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (268. - 826. i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-4.84e3 + 1.49e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.58e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.35e3 - 987. i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.87e4 + 2.08e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-5.73e3 + 1.76e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.46e4 + 4.52e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (2.58e4 + 1.87e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (1.10e3 - 803. i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.33e4 - 4.10e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-3.57e4 + 2.59e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (4.78e3 + 3.47e3i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (439. + 1.35e3i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-9.04e4 + 6.57e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−14.61153676862231681962200039962, −12.51549350937913175291497637499, −11.71532716173281683651987255694, −10.76555878547744134482830778116, −10.15848452647353555088571827227, −8.422505892173108915816735638907, −6.45107911146102608532015941280, −4.82028302030127944121493312018, −3.62961834592630909829117513316, −0.37837964083513634565555856360,
1.04717734246091800129442270573, 4.75481861727204103078455984243, 5.93972113465864771817866656802, 7.31373493853965508049136697013, 8.052921953922909444642166253680, 10.16840502640829394533529813695, 11.62344058754553553268332868819, 12.29311205398929891944401919574, 13.39161676709406185907613816495, 14.99478003668251464811786260479
