L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)9-s + (−3 + 5.19i)11-s + 2·13-s + (−2 − 3.46i)19-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s − 0.999·27-s + 6·29-s + (4 − 6.92i)31-s + (3 + 5.19i)33-s + (−1 − 1.73i)37-s + (1 − 1.73i)39-s + 12·41-s + 4·43-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.166 − 0.288i)9-s + (−0.904 + 1.56i)11-s + 0.554·13-s + (−0.458 − 0.794i)19-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s − 0.192·27-s + 1.11·29-s + (0.718 − 1.24i)31-s + (0.522 + 0.904i)33-s + (−0.164 − 0.284i)37-s + (0.160 − 0.277i)39-s + 1.87·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.806578091\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.806578091\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.778935401036831239947267251023, −8.017069253397462394286008874360, −7.43304609625792039376836339982, −6.58070181055153680044381197078, −5.93615149394874140145709010498, −4.65574677400007537791932302968, −4.24858645066785721866275906937, −2.64498024173241659152332272215, −2.27278853436506697571150202001, −0.71040944456387151011724551109,
1.04903925782933709581693766877, 2.55499019634426510586029306901, 3.37804046707988641401349354688, 4.10081317745311599115166427115, 5.32447292644377262542449593788, 5.75848343351756164234287376954, 6.73037548610876755722401835656, 7.83923337769409244194363015835, 8.373679238947412289545268438251, 8.948570264024871620121846425338