| L(s) = 1 | + (−2.43 − 0.517i)2-s + (0.0745 − 0.709i)3-s + (3.84 + 1.71i)4-s + (1.47 + 1.63i)5-s + (−0.549 + 1.69i)6-s + (0.522 + 2.59i)7-s + (−4.44 − 3.22i)8-s + (2.43 + 0.517i)9-s + (−2.74 − 4.75i)10-s + (1.49 − 2.59i)12-s + (1.01 + 3.12i)13-s + (0.0712 − 6.58i)14-s + (1.27 − 0.924i)15-s + (3.52 + 3.91i)16-s + (1.45 − 0.309i)17-s + (−5.66 − 2.52i)18-s + ⋯ |
| L(s) = 1 | + (−1.72 − 0.366i)2-s + (0.0430 − 0.409i)3-s + (1.92 + 0.855i)4-s + (0.659 + 0.732i)5-s + (−0.224 + 0.690i)6-s + (0.197 + 0.980i)7-s + (−1.57 − 1.14i)8-s + (0.812 + 0.172i)9-s + (−0.868 − 1.50i)10-s + (0.433 − 0.749i)12-s + (0.281 + 0.866i)13-s + (0.0190 − 1.76i)14-s + (0.328 − 0.238i)15-s + (0.881 + 0.978i)16-s + (0.353 − 0.0751i)17-s + (−1.33 − 0.594i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.810172 + 0.319491i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.810172 + 0.319491i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.522 - 2.59i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (2.43 + 0.517i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.0745 + 0.709i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-1.47 - 1.63i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (-1.01 - 3.12i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.45 + 0.309i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-6.31 + 2.81i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (3.24 - 5.62i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.33 - 0.969i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.57 - 1.74i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.580 - 5.52i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (9.10 + 6.61i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 + (-1.36 + 0.606i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-0.203 + 0.226i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (11.5 + 5.14i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-8.68 - 9.64i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-2.28 - 3.96i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.50 + 10.7i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.82 - 3.48i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-4.53 - 0.962i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-0.598 + 1.84i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (1.60 - 2.77i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.574 - 1.76i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.958011430484334075550091874569, −9.598337235241181435298587452350, −8.767128850962268327267971493485, −7.85219894271979948669021923016, −7.08541161254519762416679877594, −6.44283146754475410809212631895, −5.22532073532146172594703785311, −3.28463547990347802220781629561, −2.16340448559875602208189301085, −1.47289939529669636617257590229,
0.828215194779962516517059139486, 1.70462464352279751421636089663, 3.56435762954859493393252691170, 4.91977153962199710148297248841, 5.95610823992774964239911831502, 6.98561774269783555112797000538, 7.78627687787273508526705359739, 8.389021418746444930445574657137, 9.485222868124797789651325257899, 9.872111558086532086347794248305
