| L(s) = 1 | + (−1.16 + 5.11i)2-s + (−1.24 + 1.55i)3-s + (−17.5 − 8.46i)4-s + (0.849 − 1.06i)5-s + (−6.50 − 8.15i)6-s + (−18.2 + 3.22i)7-s + (37.6 − 47.1i)8-s + (5.12 + 22.4i)9-s + (4.45 + 5.58i)10-s + (4.10 − 18.0i)11-s + (34.9 − 16.8i)12-s + (−4.72 + 20.7i)13-s + (4.80 − 97.0i)14-s + (0.603 + 2.64i)15-s + (100. + 125. i)16-s + (−113. + 54.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.412 + 1.80i)2-s + (−0.238 + 0.299i)3-s + (−2.19 − 1.05i)4-s + (0.0759 − 0.0952i)5-s + (−0.442 − 0.555i)6-s + (−0.984 + 0.174i)7-s + (1.66 − 2.08i)8-s + (0.189 + 0.831i)9-s + (0.140 + 0.176i)10-s + (0.112 − 0.493i)11-s + (0.841 − 0.405i)12-s + (−0.100 + 0.442i)13-s + (0.0916 − 1.85i)14-s + (0.0103 + 0.0454i)15-s + (1.56 + 1.95i)16-s + (−1.62 + 0.780i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.251758 - 0.432074i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.251758 - 0.432074i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (18.2 - 3.22i)T \) |
| good | 2 | \( 1 + (1.16 - 5.11i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (1.24 - 1.55i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-0.849 + 1.06i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (-4.10 + 18.0i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (4.72 - 20.7i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (113. - 54.7i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + 50.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-74.0 - 35.6i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (52.6 - 25.3i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 - 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (240. - 115. i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (-153. + 192. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-225. - 282. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (31.0 - 136. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-276. - 132. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (350. + 439. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (561. - 270. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 923.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (176. + 85.0i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (13.8 + 60.7i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + 432.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (166. + 729. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-72.8 - 319. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 1.53e3T + 9.12e5T^{2} \) |
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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
−15.83670179061578837417347826931, −15.29105863551446716661456990738, −13.80864738064681508397660773077, −13.03122860293147522847780450968, −10.77524024046029139335563382588, −9.432940231815231164251567771047, −8.488893588254187354101306678692, −6.97557559882094102695322195696, −5.97472992566443157187892095041, −4.53469739120088237400425743508,
0.44826609500619063134153523109, 2.63951369049817178523612602638, 4.25578675151271142181000478655, 6.74563589095781280729263184367, 8.818682357445503583445620453891, 9.773525423789901536673820630853, 10.78670423784644199784414099230, 12.05190859570372836883666166678, 12.75923041228698119730264525296, 13.68236663184817685891658156022
