| L(s) = 1 | + (1.61 − 1.17i)2-s + (−0.927 + 2.85i)3-s + (1.23 − 3.80i)4-s + (−9.59 − 5.73i)5-s + (1.85 + 5.70i)6-s + 16.3·7-s + (−2.47 − 7.60i)8-s + (−7.28 − 5.29i)9-s + (−22.2 + 2.00i)10-s + (52.6 − 38.2i)11-s + (9.70 + 7.05i)12-s + (−61.7 − 44.8i)13-s + (26.4 − 19.2i)14-s + (25.2 − 22.0i)15-s + (−12.9 − 9.40i)16-s + (−0.0248 − 0.0764i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 − 0.475i)4-s + (−0.858 − 0.512i)5-s + (0.126 + 0.388i)6-s + 0.882·7-s + (−0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.704 + 0.0633i)10-s + (1.44 − 1.04i)11-s + (0.233 + 0.169i)12-s + (−1.31 − 0.956i)13-s + (0.504 − 0.366i)14-s + (0.434 − 0.379i)15-s + (−0.202 − 0.146i)16-s + (−0.000354 − 0.00109i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00916 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00916 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.29309 - 1.30499i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29309 - 1.30499i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.61 + 1.17i)T \) |
| 3 | \( 1 + (0.927 - 2.85i)T \) |
| 5 | \( 1 + (9.59 + 5.73i)T \) |
| good | 7 | \( 1 - 16.3T + 343T^{2} \) |
| 11 | \( 1 + (-52.6 + 38.2i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (61.7 + 44.8i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (0.0248 + 0.0764i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (32.5 + 100. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-110. + 80.1i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-16.5 + 51.0i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-71.7 - 220. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (28.2 + 20.5i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-100. - 73.0i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 354.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (103. - 319. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-21.5 + 66.1i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-473. - 343. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-70.3 + 51.1i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-304. - 935. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-121. + 375. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-246. + 179. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-410. + 1.26e3i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (22.4 + 69.0i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (158. - 115. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (280. - 862. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−12.06113016560150299557539729905, −11.44395794296530878436945542717, −10.62321531602922794462645091426, −9.186083257318429192753714303944, −8.287291122391800746838352952443, −6.79232916516171915392122691642, −5.15962635783719876407324933552, −4.46423535686434456839084514543, −3.10637400306728717529349400661, −0.800851020437518397832787938688,
1.93700546246199290578989953301, 3.90996664165465961173859937188, 4.92062727847272345778186399349, 6.64138610628825121990671658632, 7.24749537564010389914704612464, 8.235266043021047491041777191711, 9.695734084346847456271726007403, 11.34922781917058560426310045033, 11.85651471636499944673375279356, 12.58224803539727785904380028084
