| L(s) = 1 | + (1.76 + 1.28i)2-s + (−0.913 − 0.406i)3-s + (0.855 + 2.63i)4-s + (−0.272 + 0.471i)5-s + (−1.09 − 1.89i)6-s + (0.385 − 0.428i)7-s + (−0.517 + 1.59i)8-s + (0.669 + 0.743i)9-s + (−1.08 + 0.483i)10-s + (−4.22 − 0.898i)11-s + (0.289 − 2.75i)12-s + (−0.448 − 4.26i)13-s + (1.23 − 0.261i)14-s + (0.440 − 0.319i)15-s + (1.51 − 1.10i)16-s + (−0.117 + 0.0249i)17-s + ⋯ |
| L(s) = 1 | + (1.24 + 0.907i)2-s + (−0.527 − 0.234i)3-s + (0.427 + 1.31i)4-s + (−0.121 + 0.210i)5-s + (−0.445 − 0.771i)6-s + (0.145 − 0.161i)7-s + (−0.182 + 0.563i)8-s + (0.223 + 0.247i)9-s + (−0.343 + 0.152i)10-s + (−1.27 − 0.270i)11-s + (0.0834 − 0.794i)12-s + (−0.124 − 1.18i)13-s + (0.328 − 0.0699i)14-s + (0.113 − 0.0825i)15-s + (0.379 − 0.275i)16-s + (−0.0285 + 0.00606i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31016 + 0.683151i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31016 + 0.683151i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.913 + 0.406i)T \) |
| 31 | \( 1 + (3.58 + 4.26i)T \) |
| good | 2 | \( 1 + (-1.76 - 1.28i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (0.272 - 0.471i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.385 + 0.428i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (4.22 + 0.898i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.448 + 4.26i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (0.117 - 0.0249i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.0652 - 0.621i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (2.88 - 8.89i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.33 - 3.87i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (0.992 + 1.71i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.07 - 2.25i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.370 + 3.52i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-2.68 + 1.94i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.45 + 3.83i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (8.65 + 3.85i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + (-1.98 + 3.43i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.374 - 0.416i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-12.3 - 2.63i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (13.0 - 2.77i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-6.20 + 2.76i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (5.19 + 15.9i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (3.46 + 10.6i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−14.13439806608699513692794527631, −13.21267844129209865465431942670, −12.56768639845271732218892112196, −11.24942195118366065866731488387, −10.12394755058264468948940869014, −7.975413474727422875999441299808, −7.21649284958804253563254534065, −5.77286641305846392253039215281, −5.09458585623279109378405614629, −3.35599675130856435067858733497,
2.44723390741900416881963574669, 4.30950145647208132095693286736, 5.08893374366070975066702217532, 6.51992585401833581040175119776, 8.389580801879973430635084551192, 10.12267382971922357404694263488, 10.90450223134032991643288981226, 12.04344467826511508903135824301, 12.55599133297464828508184188633, 13.72329673731653485668668876866
