L(s) = 1 | + 4.18·5-s − i·7-s + 1.86i·11-s + 3.07i·13-s + 0.504i·17-s − 3.05·19-s + 5.97·23-s + 12.4·25-s + 3.52·29-s − 10.8i·31-s − 4.18i·35-s + 11.3i·37-s − 7.26i·41-s − 9.02·43-s − 0.327·47-s + ⋯ |
L(s) = 1 | + 1.87·5-s − 0.377i·7-s + 0.563i·11-s + 0.853i·13-s + 0.122i·17-s − 0.701·19-s + 1.24·23-s + 2.49·25-s + 0.655·29-s − 1.95i·31-s − 0.706i·35-s + 1.86i·37-s − 1.13i·41-s − 1.37·43-s − 0.0477·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.112617805\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.112617805\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 4.18T + 5T^{2} \) |
| 11 | \( 1 - 1.86iT - 11T^{2} \) |
| 13 | \( 1 - 3.07iT - 13T^{2} \) |
| 17 | \( 1 - 0.504iT - 17T^{2} \) |
| 19 | \( 1 + 3.05T + 19T^{2} \) |
| 23 | \( 1 - 5.97T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 + 10.8iT - 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 + 7.26iT - 41T^{2} \) |
| 43 | \( 1 + 9.02T + 43T^{2} \) |
| 47 | \( 1 + 0.327T + 47T^{2} \) |
| 53 | \( 1 - 8.32T + 53T^{2} \) |
| 59 | \( 1 - 5.98iT - 59T^{2} \) |
| 61 | \( 1 + 13.2iT - 61T^{2} \) |
| 67 | \( 1 - 9.21T + 67T^{2} \) |
| 71 | \( 1 - 9.02T + 71T^{2} \) |
| 73 | \( 1 - 0.416T + 73T^{2} \) |
| 79 | \( 1 - 6.00iT - 79T^{2} \) |
| 83 | \( 1 + 2.62iT - 83T^{2} \) |
| 89 | \( 1 - 5.69iT - 89T^{2} \) |
| 97 | \( 1 - 6.21T + 97T^{2} \) |
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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
−8.239443040045462249272806543699, −7.12511458726586738847741949662, −6.59274063385602877638117016858, −6.13829261036567448669104768500, −5.14956527225759037644690758883, −4.71476781864738296596487460751, −3.67117278076495853068717227699, −2.48795717001459160008893575243, −1.99820312079405234402000614749, −1.03715061124041856407541926892,
0.905673221027615804166094589841, 1.85225576389688377042088369419, 2.71806382229522791413823646804, 3.29328270109801243048617175519, 4.71029071148669192441176734870, 5.36892365128502216422011027566, 5.78108961553794280806465024262, 6.59850130519712909851480512686, 7.06347037510148686041574281358, 8.380680417244081919228307005133