| L(s) = 1 | + (1.06 + 0.934i)2-s + (1.92 − 2.34i)3-s + (0.253 + 1.98i)4-s + (0.956 − 0.290i)5-s + (4.23 − 0.690i)6-s + (−0.337 − 0.0670i)7-s + (−1.58 + 2.34i)8-s + (−1.20 − 6.07i)9-s + (1.28 + 0.586i)10-s + (1.43 + 0.141i)11-s + (5.13 + 3.22i)12-s + (0.936 − 3.08i)13-s + (−0.295 − 0.386i)14-s + (1.16 − 2.80i)15-s + (−3.87 + 1.00i)16-s + (0.764 + 1.84i)17-s + ⋯ |
| L(s) = 1 | + (0.750 + 0.660i)2-s + (1.11 − 1.35i)3-s + (0.126 + 0.991i)4-s + (0.427 − 0.129i)5-s + (1.72 − 0.281i)6-s + (−0.127 − 0.0253i)7-s + (−0.560 + 0.828i)8-s + (−0.402 − 2.02i)9-s + (0.406 + 0.185i)10-s + (0.433 + 0.0426i)11-s + (1.48 + 0.930i)12-s + (0.259 − 0.855i)13-s + (−0.0788 − 0.103i)14-s + (0.299 − 0.723i)15-s + (−0.967 + 0.251i)16-s + (0.185 + 0.447i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.24206 - 0.315252i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.24206 - 0.315252i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.06 - 0.934i)T \) |
| 5 | \( 1 + (-0.956 + 0.290i)T \) |
| good | 3 | \( 1 + (-1.92 + 2.34i)T + (-0.585 - 2.94i)T^{2} \) |
| 7 | \( 1 + (0.337 + 0.0670i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.43 - 0.141i)T + (10.7 + 2.14i)T^{2} \) |
| 13 | \( 1 + (-0.936 + 3.08i)T + (-10.8 - 7.22i)T^{2} \) |
| 17 | \( 1 + (-0.764 - 1.84i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (1.65 - 0.882i)T + (10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (0.491 - 0.735i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.363 - 3.68i)T + (-28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (-2.36 - 2.36i)T + 31iT^{2} \) |
| 37 | \( 1 + (5.11 - 9.56i)T + (-20.5 - 30.7i)T^{2} \) |
| 41 | \( 1 + (3.15 + 2.11i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (3.23 + 3.94i)T + (-8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (-7.46 + 3.09i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (1.31 - 13.3i)T + (-51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (4.39 + 14.4i)T + (-49.0 + 32.7i)T^{2} \) |
| 61 | \( 1 + (-3.46 - 2.84i)T + (11.9 + 59.8i)T^{2} \) |
| 67 | \( 1 + (4.76 + 3.90i)T + (13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (0.968 - 4.87i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-8.75 + 1.74i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-1.83 - 0.761i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-0.518 - 0.970i)T + (-46.1 + 69.0i)T^{2} \) |
| 89 | \( 1 + (0.0159 + 0.0238i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-4.93 - 4.93i)T + 97iT^{2} \) |
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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
−10.55578016333321528732132919034, −9.233017420914510274293598343541, −8.417495112395962054030248647585, −7.926529043113093345619124808210, −6.85890877498600762670463307270, −6.36046965153499144149118927809, −5.24721550404594531040223184946, −3.69793391374183860425261816822, −2.86696570765095419565932407441, −1.60236387101106545264534477109,
2.02094929245262267484994959233, 3.01406058585949458542422420543, 3.97567992750753678461921449501, 4.63877771423690989347139683928, 5.75293826422238715268033056357, 6.87955204558128649645300441625, 8.373490672681575489614426339039, 9.329767326229564458392475285448, 9.644402615243328687230349092196, 10.55950295563344156428098193318
