| L(s) = 1 | + (−0.418 − 1.28i)2-s + (0.669 + 0.743i)3-s + (0.137 − 0.0999i)4-s + (1.63 + 2.82i)5-s + (0.676 − 1.17i)6-s + (−0.149 − 1.41i)7-s + (−2.37 − 1.72i)8-s + (−0.104 + 0.994i)9-s + (2.95 − 3.28i)10-s + (−4.38 − 1.95i)11-s + (0.166 + 0.0353i)12-s + (3.02 − 0.643i)13-s + (−1.76 + 0.784i)14-s + (−1.00 + 3.10i)15-s + (−1.12 + 3.45i)16-s + (−3.32 + 1.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.295 − 0.909i)2-s + (0.386 + 0.429i)3-s + (0.0687 − 0.0499i)4-s + (0.730 + 1.26i)5-s + (0.276 − 0.478i)6-s + (−0.0563 − 0.536i)7-s + (−0.839 − 0.610i)8-s + (−0.0348 + 0.331i)9-s + (0.935 − 1.03i)10-s + (−1.32 − 0.589i)11-s + (0.0479 + 0.0102i)12-s + (0.840 − 0.178i)13-s + (−0.471 + 0.209i)14-s + (−0.260 + 0.802i)15-s + (−0.280 + 0.863i)16-s + (−0.806 + 0.358i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.996136 - 0.335493i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.996136 - 0.335493i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.669 - 0.743i)T \) |
| 31 | \( 1 + (4.70 + 2.98i)T \) |
| good | 2 | \( 1 + (0.418 + 1.28i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (-1.63 - 2.82i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.149 + 1.41i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (4.38 + 1.95i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-3.02 + 0.643i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (3.32 - 1.48i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (3.39 + 0.721i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-6.09 - 4.42i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.609 + 1.87i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (4.53 - 7.85i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.43 + 3.81i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-4.10 - 0.872i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (-3.27 + 10.0i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.0300 - 0.286i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-2.55 - 2.83i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + 6.98T + 61T^{2} \) |
| 67 | \( 1 + (-4.01 - 6.95i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.00 + 9.53i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-10.4 - 4.63i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (8.15 - 3.63i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (4.38 - 4.86i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (1.18 - 0.864i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-7.55 + 5.49i)T + (29.9 - 92.2i)T^{2} \) |
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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
−13.71148131282813527929522280011, −13.08860104421378506964982571353, −11.09050434161953447870526297636, −10.78222274177008527674936803974, −10.01938687272801564052800288213, −8.747829872710254178455158700064, −7.06465227329508819207392191259, −5.81507555660794770402203006825, −3.49614602833508867679782455218, −2.37769803693988156969783283435,
2.34617921864878547019447121843, 5.02937177034491663478562543967, 6.15098559914367577525878471655, 7.45665900215860044829502320571, 8.729743791624039038083971296975, 9.050479637283621332107350026340, 10.89402574593493514000317474509, 12.64089219108252621283479885535, 12.85929647217562433582494412932, 14.24408256319557829985968060524
