L(s) = 1 | + (−0.900 − 0.433i)2-s + (−0.529 − 0.361i)3-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (0.320 + 0.555i)6-s + (−1.04 + 1.81i)7-s + (−0.222 − 0.974i)8-s + (−0.945 − 2.40i)9-s + (0.955 − 0.294i)10-s + (2.03 − 2.55i)11-s + (−0.0479 − 0.639i)12-s + (1.40 + 0.434i)13-s + (1.72 − 1.17i)14-s + (0.633 − 0.0955i)15-s + (−0.222 + 0.974i)16-s + (5.16 + 4.78i)17-s + ⋯ |
L(s) = 1 | + (−0.637 − 0.306i)2-s + (−0.305 − 0.208i)3-s + (0.311 + 0.390i)4-s + (−0.327 + 0.304i)5-s + (0.130 + 0.226i)6-s + (−0.395 + 0.684i)7-s + (−0.0786 − 0.344i)8-s + (−0.315 − 0.803i)9-s + (0.302 − 0.0932i)10-s + (0.613 − 0.769i)11-s + (−0.0138 − 0.184i)12-s + (0.390 + 0.120i)13-s + (0.462 − 0.314i)14-s + (0.163 − 0.0246i)15-s + (−0.0556 + 0.243i)16-s + (1.25 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.861464 - 0.176952i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.861464 - 0.176952i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (0.733 - 0.680i)T \) |
| 43 | \( 1 + (0.341 + 6.54i)T \) |
good | 3 | \( 1 + (0.529 + 0.361i)T + (1.09 + 2.79i)T^{2} \) |
| 7 | \( 1 + (1.04 - 1.81i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.03 + 2.55i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.40 - 0.434i)T + (10.7 + 7.32i)T^{2} \) |
| 17 | \( 1 + (-5.16 - 4.78i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.237 + 0.605i)T + (-13.9 - 12.9i)T^{2} \) |
| 23 | \( 1 + (-8.69 - 1.31i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-3.54 + 2.41i)T + (10.5 - 26.9i)T^{2} \) |
| 31 | \( 1 + (0.497 + 6.63i)T + (-30.6 + 4.62i)T^{2} \) |
| 37 | \( 1 + (-0.510 - 0.884i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.46 - 1.67i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (-0.00226 - 0.00284i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-7.36 + 2.27i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (1.35 - 5.93i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (0.0659 - 0.880i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (0.443 - 1.12i)T + (-49.1 - 45.5i)T^{2} \) |
| 71 | \( 1 + (9.03 - 1.36i)T + (67.8 - 20.9i)T^{2} \) |
| 73 | \( 1 + (4.72 + 1.45i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (-5.33 + 9.24i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8.62 + 5.88i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (-5.36 - 3.65i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (9.62 - 12.0i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−11.21910227672832097782098929428, −10.23760551092182432618720781704, −9.144236310377093882708382091287, −8.633396382587306921890610270810, −7.46575930663666137701758904032, −6.36700035324326298066222069645, −5.74892705392071940271780986455, −3.80249034130553387018925285336, −2.93359988708710522955611783728, −1.01410039599863898288485944466,
1.06889619008439491975203546616, 3.07284211659667569945151923403, 4.59272810718757293496196919759, 5.45263470495139409013123389965, 6.85175178603975560767013649702, 7.43900122431995523815657559428, 8.511880907797051562225142288619, 9.440012096660642428410252223327, 10.27725771645811789136395319657, 11.02623783361375967449148994993