| L(s) = 1 | + (0.0933 − 0.589i)2-s + (0.210 − 0.107i)3-s + (3.46 + 1.12i)4-s + (−3.31 − 3.74i)5-s + (−0.0436 − 0.134i)6-s + (−7.64 + 7.64i)7-s + (2.07 − 4.06i)8-s + (−5.25 + 7.23i)9-s + (−2.51 + 1.60i)10-s + (7.33 − 5.33i)11-s + (0.851 − 0.134i)12-s + (−2.12 − 13.4i)13-s + (3.79 + 5.21i)14-s + (−1.10 − 0.433i)15-s + (9.59 + 6.96i)16-s + (0.704 + 0.359i)17-s + ⋯ |
| L(s) = 1 | + (0.0466 − 0.294i)2-s + (0.0702 − 0.0357i)3-s + (0.866 + 0.281i)4-s + (−0.662 − 0.748i)5-s + (−0.00726 − 0.0223i)6-s + (−1.09 + 1.09i)7-s + (0.258 − 0.508i)8-s + (−0.584 + 0.803i)9-s + (−0.251 + 0.160i)10-s + (0.667 − 0.484i)11-s + (0.0709 − 0.0112i)12-s + (−0.163 − 1.03i)13-s + (0.270 + 0.372i)14-s + (−0.0733 − 0.0288i)15-s + (0.599 + 0.435i)16-s + (0.0414 + 0.0211i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.961180 - 0.150197i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.961180 - 0.150197i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (3.31 + 3.74i)T \) |
| good | 2 | \( 1 + (-0.0933 + 0.589i)T + (-3.80 - 1.23i)T^{2} \) |
| 3 | \( 1 + (-0.210 + 0.107i)T + (5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (7.64 - 7.64i)T - 49iT^{2} \) |
| 11 | \( 1 + (-7.33 + 5.33i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (2.12 + 13.4i)T + (-160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-0.704 - 0.359i)T + (169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-13.5 + 4.40i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-13.5 - 2.14i)T + (503. + 163. i)T^{2} \) |
| 29 | \( 1 + (36.6 + 11.9i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (2.87 + 8.84i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-9.13 + 1.44i)T + (1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (33.7 + 24.5i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-49.4 - 49.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-9.65 - 18.9i)T + (-1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (56.8 - 28.9i)T + (1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (45.6 - 62.8i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-11.4 + 8.29i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (46.8 + 23.8i)T + (2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (0.846 - 2.60i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-60.7 - 9.62i)T + (5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (6.18 + 2.01i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-50.9 + 100. i)T + (-4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-65.5 - 90.2i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (58.6 + 115. i)T + (-5.53e3 + 7.61e3i)T^{2} \) |
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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
−16.96152979915188095756561386716, −16.09769666240733093696647589496, −15.22208196423847419876229724966, −13.13614519039271023043333356267, −12.18579257629727377375926392867, −11.12964112980578338822113454307, −9.205830289706026577852263834769, −7.74516441709470580307876965470, −5.78075400279801035007472740552, −3.08943004397070765562960858767,
3.53355627929342546950010103975, 6.52286825665631089703808696221, 7.24460917229126064318521257037, 9.590994363266218252503016200986, 11.01487077074917571094520712864, 12.11249956695271839020945151232, 14.09200552801791932925696121316, 14.97679077349777733994060915547, 16.20221033606882414873804597711, 17.10479296573722037881498859686
