| L(s) = 1 | + (0.534 − 0.238i)2-s + (1.35 − 0.287i)3-s + (−1.10 + 1.23i)4-s + (0.414 + 3.94i)5-s + (0.654 − 0.475i)6-s + (2.49 − 0.890i)7-s + (−0.661 + 2.03i)8-s + (−0.994 + 0.442i)9-s + (1.16 + 2.00i)10-s + (−1.14 + 1.98i)12-s + (1.29 + 0.938i)13-s + (1.12 − 1.06i)14-s + (1.69 + 5.21i)15-s + (−0.215 − 2.05i)16-s + (−2.15 − 0.957i)17-s + (−0.426 + 0.473i)18-s + ⋯ |
| L(s) = 1 | + (0.378 − 0.168i)2-s + (0.780 − 0.165i)3-s + (−0.554 + 0.615i)4-s + (0.185 + 1.76i)5-s + (0.267 − 0.194i)6-s + (0.941 − 0.336i)7-s + (−0.233 + 0.719i)8-s + (−0.331 + 0.147i)9-s + (0.366 + 0.635i)10-s + (−0.330 + 0.572i)12-s + (0.358 + 0.260i)13-s + (0.299 − 0.285i)14-s + (0.437 + 1.34i)15-s + (−0.0538 − 0.512i)16-s + (−0.521 − 0.232i)17-s + (−0.100 + 0.111i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0229 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0229 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50246 + 1.53728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50246 + 1.53728i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.49 + 0.890i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.534 + 0.238i)T + (1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (-1.35 + 0.287i)T + (2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (-0.414 - 3.94i)T + (-4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (-1.29 - 0.938i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.15 + 0.957i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (1.39 + 1.55i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.808 + 1.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.81 - 5.57i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.0246 + 0.234i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-3.23 - 0.688i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (1.06 - 3.28i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + (-4.32 - 4.80i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.597 + 5.68i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (4.92 - 5.47i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.595 + 5.67i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (0.659 + 1.14i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.0550 - 0.0400i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.36 + 7.07i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-5.74 + 2.55i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-4.91 + 3.56i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (5.06 - 8.77i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.9 - 8.70i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.79458361442652708187536896169, −9.408367663421867513159988104946, −8.638683633002817377306116247749, −7.79956529299763018337763749981, −7.20475054367797893591080425470, −6.13796032846463513694225621149, −4.85116342240172730446385372678, −3.81245173424714513146976682028, −2.91809178137640389392381955261, −2.21356723508766753675898613423,
0.885100068728412003984167997073, 2.14372195427080066313511547090, 3.95222322499341321471033436571, 4.54705500703036022267411471555, 5.49776412877310667201259458259, 6.01917683155040165682913309060, 7.82392750708591066148053145011, 8.532520213028387576367198446881, 8.995892004310476874506185404308, 9.591084019140954277754705280096
