L(s) = 1 | + (1.76 − 3.05i)2-s + (−3.91 − 6.77i)3-s + (−2.23 − 3.87i)4-s + (−1.03 + 1.79i)5-s − 27.6·6-s + 12.4·8-s + (−17.0 + 29.6i)9-s + (3.66 + 6.35i)10-s + (−24.5 − 42.5i)11-s + (−17.4 + 30.2i)12-s + 44.8·13-s + 16.2·15-s + (39.8 − 69.0i)16-s + (13.2 + 22.9i)17-s + (60.3 + 104. i)18-s + (38.8 − 67.3i)19-s + ⋯ |
L(s) = 1 | + (0.624 − 1.08i)2-s + (−0.752 − 1.30i)3-s + (−0.279 − 0.483i)4-s + (−0.0928 + 0.160i)5-s − 1.87·6-s + 0.551·8-s + (−0.633 + 1.09i)9-s + (0.115 + 0.200i)10-s + (−0.674 − 1.16i)11-s + (−0.420 + 0.728i)12-s + 0.956·13-s + 0.279·15-s + (0.623 − 1.07i)16-s + (0.189 + 0.327i)17-s + (0.790 + 1.36i)18-s + (0.469 − 0.812i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.330427 - 1.44048i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.330427 - 1.44048i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + (-1.76 + 3.05i)T + (-4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (3.91 + 6.77i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (1.03 - 1.79i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (24.5 + 42.5i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 44.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-13.2 - 22.9i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-38.8 + 67.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (27.8 - 48.2i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-152. - 264. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (38.5 - 66.8i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-134. + 233. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-70.5 - 122. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (212. + 367. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (293. - 509. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-89.8 - 155. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 674.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-118. - 205. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (247. - 429. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 24.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-536. + 928. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.66e3T + 9.12e5T^{2} \) |
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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
−13.75652343729182094763131966168, −13.35346883724532054351817792569, −12.23428615069081931707381728169, −11.38757199642860054900061042119, −10.59361362619897394773398486151, −8.262757975558727835956197698098, −6.83338883670394509606544857391, −5.37276842199620785742187338700, −3.14884325327222913647353029236, −1.18779700036515412361999396350,
4.20923120209997597634682106564, 5.15227549073786800362189762183, 6.33880726633345803034014811769, 7.992978741011278982974390420850, 9.854329189219458628169342635134, 10.71266171870067065809490291261, 12.18821916243627376026226051536, 13.66053270462215235324394773545, 14.90602163123814247477148539328, 15.69403105828221176338431785134