| L(s) = 1 | + (0.415 − 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (0.654 − 0.755i)6-s + (0.0903 − 0.628i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−1.92 − 4.22i)11-s + (−0.415 − 0.909i)12-s + (−0.580 − 4.03i)13-s + (−0.533 − 0.343i)14-s + (0.959 − 0.281i)15-s + (−0.142 + 0.989i)16-s + (−0.821 + 0.947i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (0.376 − 0.241i)5-s + (0.267 − 0.308i)6-s + (0.0341 − 0.237i)7-s + (−0.339 + 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.0450 − 0.313i)10-s + (−0.581 − 1.27i)11-s + (−0.119 − 0.262i)12-s + (−0.160 − 1.11i)13-s + (−0.142 − 0.0917i)14-s + (0.247 − 0.0727i)15-s + (−0.0355 + 0.247i)16-s + (−0.199 + 0.229i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31680 - 1.54956i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31680 - 1.54956i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (-4.65 + 1.14i)T \) |
| good | 7 | \( 1 + (-0.0903 + 0.628i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (1.92 + 4.22i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.580 + 4.03i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.821 - 0.947i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-0.992 - 1.14i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-3.08 + 3.55i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-4.34 + 1.27i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-8.68 - 5.58i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (7.73 - 4.96i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (7.97 + 2.34i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 5.16T + 47T^{2} \) |
| 53 | \( 1 + (-0.247 + 1.72i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.112 - 0.784i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (9.73 - 2.85i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (0.0920 - 0.201i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.51 + 5.50i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-10.6 - 12.3i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.88 - 13.1i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-0.774 - 0.497i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-2.72 - 0.799i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-5.73 + 3.68i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.24963042737912263955291401913, −9.593740163164006132203038017073, −8.410223089884747868196364652086, −8.054077617220720607219077349547, −6.51573186975083679537664647492, −5.50991623306799657152365812617, −4.64340555324352815911982873925, −3.33849061820770487243269658967, −2.64651499140812107536862473817, −0.969615477930258690204059001873,
1.95598324986966265086174153376, 3.05408234905318957663372524754, 4.47088992388582246139393995163, 5.16900617029176484969795923806, 6.55854291444246431052671859262, 7.06970715330839150731484803277, 7.953517600950078263438394373102, 9.033329000454300216822894078303, 9.557439214060182763004678174510, 10.54814619679676624536383477554
