| L(s) = 1 | + (−2.56 + 0.0517i)2-s + (−1.06 + 1.36i)3-s + (4.59 − 0.185i)4-s + (1.13 − 0.207i)5-s + (2.66 − 3.56i)6-s + (−0.888 + 0.587i)7-s + (−6.65 + 0.402i)8-s + (−0.730 − 2.90i)9-s + (−2.90 + 0.592i)10-s + (0.952 + 1.72i)11-s + (−4.63 + 6.46i)12-s + (−1.90 + 3.06i)13-s + (2.25 − 1.55i)14-s + (−0.924 + 1.77i)15-s + (7.90 − 0.638i)16-s + (0.362 − 1.77i)17-s + ⋯ |
| L(s) = 1 | + (−1.81 + 0.0365i)2-s + (−0.615 + 0.788i)3-s + (2.29 − 0.0925i)4-s + (0.507 − 0.0929i)5-s + (1.08 − 1.45i)6-s + (−0.335 + 0.222i)7-s + (−2.35 + 0.142i)8-s + (−0.243 − 0.969i)9-s + (−0.917 + 0.187i)10-s + (0.287 + 0.521i)11-s + (−1.33 + 1.86i)12-s + (−0.528 + 0.848i)13-s + (0.601 − 0.415i)14-s + (−0.238 + 0.457i)15-s + (1.97 − 0.159i)16-s + (0.0880 − 0.431i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.128724 + 0.343406i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.128724 + 0.343406i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.06 - 1.36i)T \) |
| 13 | \( 1 + (1.90 - 3.06i)T \) |
| good | 2 | \( 1 + (2.56 - 0.0517i)T + (1.99 - 0.0805i)T^{2} \) |
| 5 | \( 1 + (-1.13 + 0.207i)T + (4.67 - 1.77i)T^{2} \) |
| 7 | \( 1 + (0.888 - 0.587i)T + (2.74 - 6.43i)T^{2} \) |
| 11 | \( 1 + (-0.952 - 1.72i)T + (-5.87 + 9.29i)T^{2} \) |
| 17 | \( 1 + (-0.362 + 1.77i)T + (-15.6 - 6.66i)T^{2} \) |
| 19 | \( 1 + (-6.15 - 1.64i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.888 + 1.53i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.826 + 0.794i)T + (1.16 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-2.41 - 5.36i)T + (-20.5 + 23.2i)T^{2} \) |
| 37 | \( 1 + (0.0929 - 0.920i)T + (-36.2 - 7.40i)T^{2} \) |
| 41 | \( 1 + (-0.394 - 2.77i)T + (-39.3 + 11.4i)T^{2} \) |
| 43 | \( 1 + (5.95 + 4.86i)T + (8.60 + 42.1i)T^{2} \) |
| 47 | \( 1 + (4.05 - 12.9i)T + (-38.6 - 26.6i)T^{2} \) |
| 53 | \( 1 + (6.38 - 7.20i)T + (-6.38 - 52.6i)T^{2} \) |
| 59 | \( 1 + (-5.15 + 4.38i)T + (9.46 - 58.2i)T^{2} \) |
| 61 | \( 1 + (6.86 + 2.29i)T + (48.7 + 36.6i)T^{2} \) |
| 67 | \( 1 + (4.08 + 4.42i)T + (-5.39 + 66.7i)T^{2} \) |
| 71 | \( 1 + (8.24 + 3.31i)T + (51.2 + 49.1i)T^{2} \) |
| 73 | \( 1 + (2.02 + 1.22i)T + (33.9 + 64.6i)T^{2} \) |
| 79 | \( 1 + (0.904 - 0.474i)T + (44.8 - 65.0i)T^{2} \) |
| 83 | \( 1 + (2.37 - 3.02i)T + (-19.8 - 80.5i)T^{2} \) |
| 89 | \( 1 + (-2.81 - 10.5i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-3.68 - 10.3i)T + (-75.1 + 61.3i)T^{2} \) |
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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
−10.94422175690944611903492764357, −9.978068283792328686726457124423, −9.519543042612106790423342904005, −9.097527784625564404399265467279, −7.78017968343323667427823885861, −6.82137739664322791411154877222, −6.02802407936341172484911632471, −4.78817317097073535355833601969, −3.02649348616143264174693022942, −1.46010640085714755246973779948,
0.44092302674705273905915682896, 1.72076439723382593005403324978, 3.01525323592152700846237817267, 5.44012734439584188952165023109, 6.29635841856541434904486650417, 7.18966326243407843277889064394, 7.85129148883629875084018686640, 8.742864152277596530514765372132, 9.927609861574741108263368008216, 10.19182187592570606662063399462
