| L(s) = 1 | + (−2.04 − 1.39i)2-s + (−2.21 − 2.05i)3-s + (1.50 + 3.84i)4-s + (1.22 − 0.379i)5-s + (1.66 + 7.28i)6-s + (2.38 − 1.15i)7-s + (1.17 − 5.14i)8-s + (0.456 + 6.09i)9-s + (−3.04 − 0.938i)10-s + (−0.329 + 4.39i)11-s + (4.55 − 11.6i)12-s + (0.900 − 0.433i)13-s + (−6.48 − 0.967i)14-s + (−3.49 − 1.68i)15-s + (−3.52 + 3.27i)16-s + (3.84 − 0.579i)17-s + ⋯ |
| L(s) = 1 | + (−1.44 − 0.986i)2-s + (−1.27 − 1.18i)3-s + (0.754 + 1.92i)4-s + (0.549 − 0.169i)5-s + (0.678 + 2.97i)6-s + (0.900 − 0.435i)7-s + (0.415 − 1.82i)8-s + (0.152 + 2.03i)9-s + (−0.962 − 0.296i)10-s + (−0.0992 + 1.32i)11-s + (1.31 − 3.35i)12-s + (0.249 − 0.120i)13-s + (−1.73 − 0.258i)14-s + (−0.902 − 0.434i)15-s + (−0.881 + 0.818i)16-s + (0.932 − 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.123977 + 0.289502i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.123977 + 0.289502i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.38 + 1.15i)T \) |
| 13 | \( 1 + (-0.900 + 0.433i)T \) |
| good | 2 | \( 1 + (2.04 + 1.39i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (2.21 + 2.05i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-1.22 + 0.379i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.329 - 4.39i)T + (-10.8 - 1.63i)T^{2} \) |
| 17 | \( 1 + (-3.84 + 0.579i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (4.06 + 7.04i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.13 + 1.07i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (3.99 + 5.00i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.34 + 2.32i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.33 + 5.94i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (0.600 - 2.62i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.945 - 4.14i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (6.58 + 4.48i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (1.24 + 3.18i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (3.84 + 1.18i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (1.19 - 3.03i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-5.21 + 9.02i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.56 - 8.22i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-7.23 + 4.93i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (7.58 + 13.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.33 - 0.642i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.376 - 5.02i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 2.98T + 97T^{2} \) |
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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
−10.15013912420597719100355276849, −9.464335565006862892711697090148, −8.091717248473094564186642908267, −7.63645783036839609905560525879, −6.78760522468308070739914033384, −5.64026221804345537368452521190, −4.43876410205397838830464042482, −2.19393592184776388158888439238, −1.64945101748396437673685280303, −0.34079199244295069848377189570,
1.47202069231513403667857822286, 3.86626385955109721432371732979, 5.36345493137414344928979251076, 5.87409626807977702903762728972, 6.33804615485707599813116478748, 7.934442803485953928623435233949, 8.483479146342945369159599768927, 9.506267192580780352833233061168, 10.22478858962905807352109230340, 10.67309192777305820321376269111
