L(s) = 1 | + (0.228 − 0.0611i)2-s + (−0.258 + 0.965i)3-s + (−1.68 + 0.972i)4-s + (1.18 − 1.89i)5-s + 0.236i·6-s + (−0.658 + 0.658i)8-s + (−0.866 − 0.499i)9-s + (0.155 − 0.504i)10-s + (−1.99 − 3.45i)11-s + (−0.503 − 1.87i)12-s + (−0.500 − 0.500i)13-s + (1.52 + 1.63i)15-s + (1.83 − 3.17i)16-s + (−2.29 − 0.614i)17-s + (−0.228 − 0.0611i)18-s + (−3.60 + 6.25i)19-s + ⋯ |
L(s) = 1 | + (0.161 − 0.0432i)2-s + (−0.149 + 0.557i)3-s + (−0.841 + 0.486i)4-s + (0.532 − 0.846i)5-s + 0.0964i·6-s + (−0.232 + 0.232i)8-s + (−0.288 − 0.166i)9-s + (0.0492 − 0.159i)10-s + (−0.600 − 1.04i)11-s + (−0.145 − 0.542i)12-s + (−0.138 − 0.138i)13-s + (0.392 + 0.423i)15-s + (0.458 − 0.794i)16-s + (−0.556 − 0.148i)17-s + (−0.0537 − 0.0144i)18-s + (−0.828 + 1.43i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.454756 - 0.559518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.454756 - 0.559518i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 + (-1.18 + 1.89i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.228 + 0.0611i)T + (1.73 - i)T^{2} \) |
| 11 | \( 1 + (1.99 + 3.45i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.500 + 0.500i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.29 + 0.614i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (3.60 - 6.25i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.88 + 7.04i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 3.65iT - 29T^{2} \) |
| 31 | \( 1 + (-4.27 + 2.46i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.399 - 0.106i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 7.63iT - 41T^{2} \) |
| 43 | \( 1 + (-3.65 + 3.65i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.111 - 0.417i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (7.37 + 1.97i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.05 - 5.29i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.15 + 3.55i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.345 + 1.28i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 + (0.506 - 1.88i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (7.48 + 4.32i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.9 - 11.9i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.91 + 6.77i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.43 - 7.43i)T - 97iT^{2} \) |
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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
−10.10603680068818609851938044694, −9.220609025984694240131765888893, −8.436744652538014191336790320567, −8.033186692534461320237035644981, −6.23503664924017338998929313899, −5.53081156637520000568854517257, −4.58537545772291000324623507790, −3.86750807093778786467687380580, −2.48361659766354746164795447027, −0.35589677643201921255165083199,
1.71056851791872527573965417141, 2.90050104633489717067596054733, 4.42464146699763977946099151433, 5.24901261896262107999124573529, 6.28493274387596965283076977104, 6.96485735818316747317072090149, 7.933811972907351313471936402327, 9.107490594847177330544834886991, 9.747833701680530950363992097296, 10.59210416437660280234017889690