| L(s) = 1 | + (−0.618 + 1.90i)2-s + (−1.87 − 1.36i)3-s + (−3.23 − 2.35i)4-s + (−9.42 − 6.01i)5-s + (3.75 − 2.72i)6-s − 15.5·7-s + (6.47 − 4.70i)8-s + (−6.68 − 20.5i)9-s + (17.2 − 14.2i)10-s + (1.43 − 4.40i)11-s + (2.86 + 8.81i)12-s + (−0.992 − 3.05i)13-s + (9.62 − 29.6i)14-s + (9.47 + 24.1i)15-s + (4.94 + 15.2i)16-s + (−74.0 + 53.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.360 − 0.262i)3-s + (−0.404 − 0.293i)4-s + (−0.842 − 0.538i)5-s + (0.255 − 0.185i)6-s − 0.840·7-s + (0.286 − 0.207i)8-s + (−0.247 − 0.761i)9-s + (0.546 − 0.449i)10-s + (0.0392 − 0.120i)11-s + (0.0689 + 0.212i)12-s + (−0.0211 − 0.0651i)13-s + (0.183 − 0.565i)14-s + (0.163 + 0.415i)15-s + (0.0772 + 0.237i)16-s + (−1.05 + 0.767i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.152308 - 0.278966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.152308 - 0.278966i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.618 - 1.90i)T \) |
| 5 | \( 1 + (9.42 + 6.01i)T \) |
| good | 3 | \( 1 + (1.87 + 1.36i)T + (8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + 15.5T + 343T^{2} \) |
| 11 | \( 1 + (-1.43 + 4.40i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (0.992 + 3.05i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (74.0 - 53.8i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-17.0 + 12.4i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (15.8 - 48.6i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (146. + 106. i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-260. + 189. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (53.3 + 164. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (126. + 389. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 328.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (360. + 261. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (63.8 + 46.4i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-218. - 673. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-44.5 + 137. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-367. + 266. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-103. - 75.3i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (36.9 - 113. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (321. + 233. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (709. - 515. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (115. - 353. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (1.34e3 + 976. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−15.10515211723839561848463405054, −13.43109057314039914456964029154, −12.45131438355707322750434198319, −11.29121332949232976589382133605, −9.568975999767961285767526823302, −8.464544748194337542290417727791, −7.03996319521893446301108821733, −5.84737674125152058215463305549, −3.95143316121838899406263205085, −0.26278396320580851392522082005,
2.95612246289628606564052315995, 4.64448910035916490210669443605, 6.73399488675047218760267771269, 8.240353418865320361667207424851, 9.766144186314812314581883928908, 10.88269417014991758295661466619, 11.69868061776754603030090298188, 12.95483487563321380858332239783, 14.19504499485595463558965549872, 15.70870955408031005887397513591
