| L(s) = 1 | + (−1.15 − 0.785i)2-s + (1.86 + 1.73i)3-s + (−0.0200 − 0.0509i)4-s + (−1.72 + 0.532i)5-s + (−0.789 − 3.46i)6-s + (−0.964 − 2.46i)7-s + (−0.637 + 2.79i)8-s + (0.259 + 3.46i)9-s + (2.40 + 0.743i)10-s + (0.185 − 2.48i)11-s + (0.0509 − 0.129i)12-s + (0.900 − 0.433i)13-s + (−0.824 + 3.59i)14-s + (−4.14 − 1.99i)15-s + (2.84 − 2.64i)16-s + (7.11 − 1.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.814 − 0.555i)2-s + (1.07 + 0.999i)3-s + (−0.0100 − 0.0254i)4-s + (−0.772 + 0.238i)5-s + (−0.322 − 1.41i)6-s + (−0.364 − 0.931i)7-s + (−0.225 + 0.987i)8-s + (0.0866 + 1.15i)9-s + (0.762 + 0.235i)10-s + (0.0560 − 0.747i)11-s + (0.0147 − 0.0374i)12-s + (0.249 − 0.120i)13-s + (−0.220 + 0.961i)14-s + (−1.07 − 0.515i)15-s + (0.712 − 0.661i)16-s + (1.72 − 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.968672 - 0.506533i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968672 - 0.506533i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.964 + 2.46i)T \) |
| 13 | \( 1 + (-0.900 + 0.433i)T \) |
| good | 2 | \( 1 + (1.15 + 0.785i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (-1.86 - 1.73i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (1.72 - 0.532i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.185 + 2.48i)T + (-10.8 - 1.63i)T^{2} \) |
| 17 | \( 1 + (-7.11 + 1.07i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (1.55 + 2.69i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-8.21 - 1.23i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (5.32 + 6.67i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-4.83 + 8.37i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.590 - 1.50i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (0.591 - 2.59i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-1.01 - 4.44i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-6.50 - 4.43i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (1.64 + 4.19i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-9.86 - 3.04i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-2.14 + 5.47i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-1.07 + 1.86i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.825 - 1.03i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (6.46 - 4.40i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (5.70 + 9.88i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (14.3 + 6.89i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.638 + 8.51i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−10.10587199276157693712267987077, −9.746035380579231470316656949241, −8.916057119074951200381813582550, −8.063032072580431160996202359148, −7.44860012797829854587122048465, −5.83167680899834229743100364811, −4.49102060952965724072941321548, −3.51787767316373606255172197536, −2.83084109471752752688970885066, −0.803471105647484720192474961251,
1.34368541755597349816129430778, 2.92049846025769895868427720703, 3.78936384257495246344041734005, 5.48714848123817802236597496790, 6.91184024286002761730091319225, 7.27522575959335180875423261415, 8.279413771750021691894879567959, 8.608185512507062052758160481868, 9.366507849002180069668523431385, 10.37092117298124091672176970889
