| L(s) = 1 | + (−1.66 − 0.778i)2-s + (0.823 + 1.17i)3-s + (0.896 + 1.06i)4-s + (−0.649 − 2.13i)5-s + (−0.459 − 2.60i)6-s + (4.27 + 1.14i)7-s + (0.288 + 1.07i)8-s + (0.320 − 0.879i)9-s + (−0.581 + 4.07i)10-s + (1.77 − 3.07i)11-s + (−0.518 + 1.93i)12-s + (−1.25 − 0.878i)13-s + (−6.24 − 5.23i)14-s + (1.98 − 2.52i)15-s + (0.841 − 4.77i)16-s + (−2.08 + 4.46i)17-s + ⋯ |
| L(s) = 1 | + (−1.18 − 0.550i)2-s + (0.475 + 0.679i)3-s + (0.448 + 0.533i)4-s + (−0.290 − 0.956i)5-s + (−0.187 − 1.06i)6-s + (1.61 + 0.432i)7-s + (0.102 + 0.381i)8-s + (0.106 − 0.293i)9-s + (−0.183 + 1.28i)10-s + (0.535 − 0.926i)11-s + (−0.149 + 0.558i)12-s + (−0.347 − 0.243i)13-s + (−1.66 − 1.40i)14-s + (0.511 − 0.652i)15-s + (0.210 − 1.19i)16-s + (−0.504 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.671317 - 0.210321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.671317 - 0.210321i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.649 + 2.13i)T \) |
| 19 | \( 1 + (0.520 - 4.32i)T \) |
| good | 2 | \( 1 + (1.66 + 0.778i)T + (1.28 + 1.53i)T^{2} \) |
| 3 | \( 1 + (-0.823 - 1.17i)T + (-1.02 + 2.81i)T^{2} \) |
| 7 | \( 1 + (-4.27 - 1.14i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.77 + 3.07i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.25 + 0.878i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (2.08 - 4.46i)T + (-10.9 - 13.0i)T^{2} \) |
| 23 | \( 1 + (2.61 + 0.228i)T + (22.6 + 3.99i)T^{2} \) |
| 29 | \( 1 + (2.05 + 0.748i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (5.90 - 3.40i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.62 + 1.62i)T + 37iT^{2} \) |
| 41 | \( 1 + (-3.31 - 0.584i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.476 + 5.44i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (5.62 - 2.62i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (0.296 - 3.38i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (1.77 - 0.647i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.98 + 6.70i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.21 - 2.61i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-2.76 + 3.29i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (10.7 - 7.53i)T + (24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (0.548 - 3.11i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.251 + 0.939i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.53 - 8.72i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (4.20 + 1.96i)T + (62.3 + 74.3i)T^{2} \) |
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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
−14.21949282344722394538751373727, −12.42244073008376209634073473139, −11.48573709718931700501894948821, −10.55850878749627012007454903501, −9.303120524887875762378100327260, −8.550828637131909792232811118442, −8.028769318763869012634058149047, −5.44160581585304599252641321012, −4.03172815423675912160017737568, −1.62243142417171623991984933948,
1.97244119400511705972139224254, 4.50372201853321791739343790414, 7.00331692063175179610205668671, 7.35694929910673799292305630691, 8.258151486128146793401448829605, 9.510928115488922217044984426504, 10.77050303878050225364729652302, 11.66449592027111335917161282476, 13.28509778800743868104996659799, 14.36067825803210644229796602327
