L(s) = 1 | + (0.179 − 1.40i)2-s + i·3-s + (−1.93 − 0.502i)4-s − i·5-s + (1.40 + 0.179i)6-s + 1.34·7-s + (−1.05 + 2.62i)8-s − 9-s + (−1.40 − 0.179i)10-s − 0.705·11-s + (0.502 − 1.93i)12-s − 3.25·13-s + (0.240 − 1.88i)14-s + 15-s + (3.49 + 1.94i)16-s + 4.04i·17-s + ⋯ |
L(s) = 1 | + (0.126 − 0.991i)2-s + 0.577i·3-s + (−0.967 − 0.251i)4-s − 0.447i·5-s + (0.572 + 0.0730i)6-s + 0.507·7-s + (−0.371 + 0.928i)8-s − 0.333·9-s + (−0.443 − 0.0566i)10-s − 0.212·11-s + (0.144 − 0.558i)12-s − 0.903·13-s + (0.0642 − 0.503i)14-s + 0.258·15-s + (0.873 + 0.486i)16-s + 0.981i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2435991152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2435991152\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.179 + 1.40i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (2.51 + 4.08i)T \) |
good | 7 | \( 1 - 1.34T + 7T^{2} \) |
| 11 | \( 1 + 0.705T + 11T^{2} \) |
| 13 | \( 1 + 3.25T + 13T^{2} \) |
| 17 | \( 1 - 4.04iT - 17T^{2} \) |
| 19 | \( 1 + 3.68T + 19T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 + 1.38iT - 31T^{2} \) |
| 37 | \( 1 - 8.57iT - 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 + 3.29iT - 47T^{2} \) |
| 53 | \( 1 - 7.62iT - 53T^{2} \) |
| 59 | \( 1 + 6.68iT - 59T^{2} \) |
| 61 | \( 1 - 9.90iT - 61T^{2} \) |
| 67 | \( 1 + 6.40T + 67T^{2} \) |
| 71 | \( 1 - 8.64iT - 71T^{2} \) |
| 73 | \( 1 + 6.69T + 73T^{2} \) |
| 79 | \( 1 + 2.03T + 79T^{2} \) |
| 83 | \( 1 + 4.44T + 83T^{2} \) |
| 89 | \( 1 + 7.34iT - 89T^{2} \) |
| 97 | \( 1 - 17.2iT - 97T^{2} \) |
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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
−10.08100654761155206332088411132, −9.120200127243566632207181634455, −8.461045317832417833365983725938, −7.76278341123787318563098359264, −6.25255779870465129514201630669, −5.31408701435536043478979739041, −4.55424588454950009969819901029, −3.92526615210244603164389014670, −2.66835114358352937231147524761, −1.66100047360206932881057523737,
0.090983841197747590509844375133, 2.00082065862988583689238348285, 3.27590548729893234589614143440, 4.45528486449283491631382668946, 5.33410338401193902975110335821, 6.08722229248938791721289237856, 7.12824390133333674850817157389, 7.48016469722790900683843252731, 8.252767796623195392074429299435, 9.208870280936906601336143404602