| L(s) = 1 | + (0.290 + 1.31i)2-s + (−2.99 + 0.0742i)3-s + (1.97 − 0.913i)4-s + (−6.07 − 1.68i)5-s + (−0.968 − 3.93i)6-s + (0.563 − 0.534i)7-s + (5.04 + 6.64i)8-s + (8.98 − 0.445i)9-s + (0.461 − 8.50i)10-s + (13.5 − 11.4i)11-s + (−5.85 + 2.88i)12-s + (4.18 − 1.41i)13-s + (0.868 + 0.588i)14-s + (18.3 + 4.60i)15-s + (−1.66 + 1.95i)16-s + (17.2 − 18.2i)17-s + ⋯ |
| L(s) = 1 | + (0.145 + 0.659i)2-s + (−0.999 + 0.0247i)3-s + (0.493 − 0.228i)4-s + (−1.21 − 0.337i)5-s + (−0.161 − 0.655i)6-s + (0.0805 − 0.0762i)7-s + (0.631 + 0.830i)8-s + (0.998 − 0.0495i)9-s + (0.0461 − 0.850i)10-s + (1.22 − 1.04i)11-s + (−0.487 + 0.240i)12-s + (0.322 − 0.108i)13-s + (0.0620 + 0.0420i)14-s + (1.22 + 0.307i)15-s + (−0.103 + 0.122i)16-s + (1.01 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.21064 - 0.101264i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21064 - 0.101264i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.99 - 0.0742i)T \) |
| 59 | \( 1 + (-26.0 + 52.9i)T \) |
| good | 2 | \( 1 + (-0.290 - 1.31i)T + (-3.63 + 1.67i)T^{2} \) |
| 5 | \( 1 + (6.07 + 1.68i)T + (21.4 + 12.8i)T^{2} \) |
| 7 | \( 1 + (-0.563 + 0.534i)T + (2.65 - 48.9i)T^{2} \) |
| 11 | \( 1 + (-13.5 + 11.4i)T + (19.5 - 119. i)T^{2} \) |
| 13 | \( 1 + (-4.18 + 1.41i)T + (134. - 102. i)T^{2} \) |
| 17 | \( 1 + (-17.2 + 18.2i)T + (-15.6 - 288. i)T^{2} \) |
| 19 | \( 1 + (6.12 + 15.3i)T + (-262. + 248. i)T^{2} \) |
| 23 | \( 1 + (-3.72 - 34.2i)T + (-516. + 113. i)T^{2} \) |
| 29 | \( 1 + (-0.148 + 0.674i)T + (-763. - 353. i)T^{2} \) |
| 31 | \( 1 + (-12.1 + 30.4i)T + (-697. - 660. i)T^{2} \) |
| 37 | \( 1 + (18.1 + 13.7i)T + (366. + 1.31e3i)T^{2} \) |
| 41 | \( 1 + (-2.30 + 21.1i)T + (-1.64e3 - 361. i)T^{2} \) |
| 43 | \( 1 + (-14.4 + 16.9i)T + (-299. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (37.0 - 10.2i)T + (1.89e3 - 1.13e3i)T^{2} \) |
| 53 | \( 1 + (-5.29 + 0.287i)T + (2.79e3 - 303. i)T^{2} \) |
| 61 | \( 1 + (22.7 - 5.00i)T + (3.37e3 - 1.56e3i)T^{2} \) |
| 67 | \( 1 + (-58.7 + 44.6i)T + (1.20e3 - 4.32e3i)T^{2} \) |
| 71 | \( 1 + (15.8 - 4.38i)T + (4.31e3 - 2.59e3i)T^{2} \) |
| 73 | \( 1 + (58.2 - 85.9i)T + (-1.97e3 - 4.95e3i)T^{2} \) |
| 79 | \( 1 + (-0.171 - 1.04i)T + (-5.91e3 + 1.99e3i)T^{2} \) |
| 83 | \( 1 + (-134. + 71.3i)T + (3.86e3 - 5.70e3i)T^{2} \) |
| 89 | \( 1 + (10.0 - 45.5i)T + (-7.18e3 - 3.32e3i)T^{2} \) |
| 97 | \( 1 + (-19.1 - 28.1i)T + (-3.48e3 + 8.74e3i)T^{2} \) |
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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
−11.97273109638804416257380110936, −11.55629610081933802485995953035, −10.89608362943300515672500239142, −9.366220704365215888239326038096, −7.943593185081525483084661371638, −7.12435197591964357454409657703, −6.07804887969548590300617452523, −5.06092473577016367706110263201, −3.73524775238660548396876565663, −0.923957767293303707712921311432,
1.48805035859349863633797998638, 3.62609831675606335183472662842, 4.43154014572385205481756080162, 6.34949265981797327239722181752, 7.09438481509828483054800319480, 8.209028515045147563386970228409, 10.04763292091075167018093712457, 10.71969258769642704655462901251, 11.74550739932307167340888590485, 12.14554664748520105541745708806
