| L(s) = 1 | + (1.46 − 1.36i)2-s + (−5.45 + 0.822i)3-s + (0.298 − 3.98i)4-s + (1.52 − 3.87i)5-s + (−6.88 + 8.62i)6-s + (−11.3 + 14.6i)7-s + (−4.98 − 6.25i)8-s + (3.29 − 1.01i)9-s + (−3.04 − 7.74i)10-s + (−56.7 − 17.5i)11-s + (1.64 + 22.0i)12-s + (17.1 + 74.9i)13-s + (3.24 + 36.8i)14-s + (−5.10 + 22.3i)15-s + (−15.8 − 2.38i)16-s + (−39.9 + 27.2i)17-s + ⋯ |
| L(s) = 1 | + (0.518 − 0.480i)2-s + (−1.05 + 0.158i)3-s + (0.0373 − 0.498i)4-s + (0.135 − 0.346i)5-s + (−0.468 + 0.587i)6-s + (−0.613 + 0.789i)7-s + (−0.220 − 0.276i)8-s + (0.122 − 0.0376i)9-s + (−0.0961 − 0.244i)10-s + (−1.55 − 0.479i)11-s + (0.0396 + 0.529i)12-s + (0.365 + 1.59i)13-s + (0.0619 + 0.704i)14-s + (−0.0879 + 0.385i)15-s + (−0.247 − 0.0372i)16-s + (−0.569 + 0.388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.628i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.778 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0338303 + 0.0957466i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0338303 + 0.0957466i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.46 + 1.36i)T \) |
| 7 | \( 1 + (11.3 - 14.6i)T \) |
| good | 3 | \( 1 + (5.45 - 0.822i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (-1.52 + 3.87i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (56.7 + 17.5i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (-17.1 - 74.9i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (39.9 - 27.2i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (37.4 + 64.9i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (76.0 + 51.8i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (119. + 57.5i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (-135. + 234. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (16.2 + 217. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (-145. - 182. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (237. - 298. i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (241. - 223. i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (-48.7 + 649. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (-29.2 - 74.4i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (-44.6 - 595. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (117. - 202. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (643. - 309. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-284. - 263. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (-113. - 196. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-61.6 + 270. i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (1.10e3 - 341. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 - 149.T + 9.12e5T^{2} \) |
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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
−13.39250056319773375408919775531, −12.83002810430529743746348360171, −11.56832811062310152674227005036, −11.05480070242554793280028901461, −9.761535818351474109741672690001, −8.559867043642464807434517639040, −6.47007788567168409211219644129, −5.63633703565733450705453513175, −4.48954146536704891926458790009, −2.46494497286512333640298849363,
0.05169950647421673874637176172, 3.13633048968520359356838211600, 4.98615201991772107425830269945, 5.97375925409657786004871142685, 7.04030384040066790356949410383, 8.177316624015747297730880892959, 10.30063955831015627567147563056, 10.70524208260922558985700648602, 12.25691259232970265017006740533, 12.97766639238117238748254124998
