| L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.587 + 1.62i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (−0.0555 + 1.73i)6-s + (−0.309 − 2.94i)7-s + (0.587 − 0.809i)8-s + (−2.30 − 1.91i)9-s + (−0.669 + 0.743i)10-s + (−0.630 − 0.280i)11-s + (0.482 + 1.66i)12-s + (−0.960 − 4.51i)13-s + (−1.20 − 2.70i)14-s + (−0.305 − 1.70i)15-s + (0.309 − 0.951i)16-s + (2.12 − 0.948i)17-s + ⋯ |
| L(s) = 1 | + (0.672 − 0.218i)2-s + (−0.339 + 0.940i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (−0.0226 + 0.706i)6-s + (−0.117 − 1.11i)7-s + (0.207 − 0.286i)8-s + (−0.769 − 0.638i)9-s + (−0.211 + 0.235i)10-s + (−0.190 − 0.0846i)11-s + (0.139 + 0.480i)12-s + (−0.266 − 1.25i)13-s + (−0.321 − 0.723i)14-s + (−0.0789 − 0.440i)15-s + (0.0772 − 0.237i)16-s + (0.516 − 0.229i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11943 - 0.919401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11943 - 0.919401i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (0.587 - 1.62i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (5.16 + 2.06i)T \) |
| good | 7 | \( 1 + (0.309 + 2.94i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (0.630 + 0.280i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (0.960 + 4.51i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-2.12 + 0.948i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (0.844 + 0.179i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (0.00251 + 0.00182i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.51 + 7.72i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.876 - 0.506i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.95 - 2.66i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (0.490 - 2.30i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-10.3 - 3.37i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.465 - 4.43i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-6.90 + 6.22i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 4.47iT - 61T^{2} \) |
| 67 | \( 1 + (6.17 + 10.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.89 + 0.199i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (0.242 - 0.545i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (3.21 + 7.22i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (6.89 - 7.65i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (2.82 - 2.05i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (8.31 - 6.04i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.14408583815872016509919105586, −9.412126385834694284146744949729, −8.012913029818593853856193447169, −7.35543777012202507340192560450, −6.15624206125382989669942984064, −5.38989065709383140873806653885, −4.37405070748042248228848653906, −3.70109510884648622901051186687, −2.78418887110293793651076770191, −0.56006393960937342996588721330,
1.71946873630947068195785267197, 2.73840459854727998553472500230, 4.06804636133324798298989305564, 5.30601571499710100577404727776, 5.77219733076761780472282223127, 6.91094983761798135516372715647, 7.38885955276028221862853600784, 8.568978720577443352782687004551, 9.050644414596962369920759168707, 10.50620043003733570451769626751
