L(s) = 1 | + (−0.5 − 0.866i)3-s + (1 − 1.73i)5-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)11-s + 4·13-s − 1.99·15-s + (3 + 5.19i)17-s + (−4 + 6.92i)19-s + (−3 + 5.19i)23-s + (0.500 + 0.866i)25-s + 0.999·27-s − 10·29-s + (−2 − 3.46i)31-s + (0.999 − 1.73i)33-s + (−3 + 5.19i)37-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.447 − 0.774i)5-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s + 1.10·13-s − 0.516·15-s + (0.727 + 1.26i)17-s + (−0.917 + 1.58i)19-s + (−0.625 + 1.08i)23-s + (0.100 + 0.173i)25-s + 0.192·27-s − 1.85·29-s + (−0.359 − 0.622i)31-s + (0.174 − 0.301i)33-s + (−0.493 + 0.854i)37-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
\(L(1)\) |
\(\approx\) |
\(1.405903075\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.405903075\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−9.124295105806833773727832379308, −8.118968934213075201095250999136, −7.82873433281682297105215498016, −6.60911955736818105125185209772, −5.82846772557277040098980208945, −5.52085651025831516810362680221, −4.16078704028136684001010309938, −3.56394001544549904949620868212, −1.72401889027666033798574163309, −1.50404947369174956930445069323,
0.49420185314976220282805971217, 2.14189477527070864479060436619, 3.13378328013269600317980961030, 3.92702906281358700852487197379, 4.96992830920037866504649465723, 5.78436641554897193390401523661, 6.53097058987306716963352900002, 7.08913600098806334463234592185, 8.238293653941586042831929169610, 9.026924650502198397311794069882