| L(s) = 1 | + (0.701 + 1.78i)2-s + (−0.192 − 2.56i)3-s + (3.16 − 2.93i)4-s + (9.37 − 6.39i)5-s + (4.44 − 2.14i)6-s + (−18.5 + 0.408i)7-s + (21.2 + 10.2i)8-s + (20.1 − 3.03i)9-s + (17.9 + 12.2i)10-s + (20.8 + 3.14i)11-s + (−8.13 − 7.55i)12-s + (−41.2 + 51.7i)13-s + (−13.7 − 32.7i)14-s + (−18.1 − 22.8i)15-s + (−0.809 + 10.8i)16-s + (−89.9 + 27.7i)17-s + ⋯ |
| L(s) = 1 | + (0.247 + 0.631i)2-s + (−0.0369 − 0.493i)3-s + (0.395 − 0.367i)4-s + (0.838 − 0.571i)5-s + (0.302 − 0.145i)6-s + (−0.999 + 0.0220i)7-s + (0.941 + 0.453i)8-s + (0.746 − 0.112i)9-s + (0.569 + 0.387i)10-s + (0.572 + 0.0862i)11-s + (−0.195 − 0.181i)12-s + (−0.880 + 1.10i)13-s + (−0.261 − 0.626i)14-s + (−0.313 − 0.392i)15-s + (−0.0126 + 0.168i)16-s + (−1.28 + 0.395i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0839i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.78695 - 0.0751484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78695 - 0.0751484i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (18.5 - 0.408i)T \) |
| good | 2 | \( 1 + (-0.701 - 1.78i)T + (-5.86 + 5.44i)T^{2} \) |
| 3 | \( 1 + (0.192 + 2.56i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (-9.37 + 6.39i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (-20.8 - 3.14i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (41.2 - 51.7i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (89.9 - 27.7i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-52.5 + 91.0i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (105. + 32.5i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (55.1 - 241. i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-82.0 - 142. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-255. - 236. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (299. + 144. i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (169. - 81.4i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (67.9 + 173. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-294. + 272. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-79.7 - 54.3i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (66.8 + 62.0i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (297. + 515. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (117. + 514. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-127. + 324. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (44.7 - 77.4i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (213. + 267. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (451. - 68.0i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 - 1.63e3T + 9.12e5T^{2} \) |
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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
−15.17620159219827379005894842809, −13.83515429640874005555918907902, −13.10915364133083068438163611285, −11.84108074634920649157755624028, −10.11472210416177293036477448685, −9.118677404708059045091448868453, −6.97939092937879767269564292741, −6.46575540341102263381141304336, −4.79043815786656392341160477337, −1.81732020230986005125096780199,
2.45962147318069270227788345559, 3.99161260145801198312503802191, 6.14495424897773148362910808607, 7.48444831745724390649363937916, 9.837891832010820193358049401431, 10.12965670789099330805089934486, 11.62798167617478182793700416354, 12.84762654772749316394309122223, 13.67819647296793985889881779360, 15.24134677065894571365499300868
