L(s) = 1 | + (1.37 + 0.336i)2-s + (1.77 + 0.924i)4-s − 3.04i·5-s − 2.23i·7-s + (2.12 + 1.86i)8-s + (1.02 − 4.17i)10-s + 2.29·11-s + 3.09·13-s + (0.752 − 3.07i)14-s + (2.29 + 3.27i)16-s − 5.58·17-s + (−4.29 − 0.758i)19-s + (2.81 − 5.39i)20-s + (3.14 + 0.771i)22-s − 5.51i·23-s + ⋯ |
L(s) = 1 | + (0.971 + 0.237i)2-s + (0.886 + 0.462i)4-s − 1.36i·5-s − 0.845i·7-s + (0.751 + 0.659i)8-s + (0.323 − 1.32i)10-s + 0.691·11-s + 0.859·13-s + (0.201 − 0.820i)14-s + (0.573 + 0.819i)16-s − 1.35·17-s + (−0.984 − 0.174i)19-s + (0.628 − 1.20i)20-s + (0.671 + 0.164i)22-s − 1.14i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.200214601\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.200214601\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 - 0.336i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4.29 + 0.758i)T \) |
good | 5 | \( 1 + 3.04iT - 5T^{2} \) |
| 7 | \( 1 + 2.23iT - 7T^{2} \) |
| 11 | \( 1 - 2.29T + 11T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 17 | \( 1 + 5.58T + 17T^{2} \) |
| 23 | \( 1 + 5.51iT - 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 - 4.25T + 31T^{2} \) |
| 37 | \( 1 + 1.24T + 37T^{2} \) |
| 41 | \( 1 + 8.33iT - 41T^{2} \) |
| 43 | \( 1 - 4.87T + 43T^{2} \) |
| 47 | \( 1 - 9.36iT - 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 2.49iT - 59T^{2} \) |
| 61 | \( 1 + 2.26iT - 61T^{2} \) |
| 67 | \( 1 - 4.49iT - 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 1.51T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 - 5.29iT - 89T^{2} \) |
| 97 | \( 1 - 1.17iT - 97T^{2} \) |
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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
−9.217211472018088136548812223157, −8.591299711872111067422649714921, −7.88598687375058977362078772363, −6.64012034332506175727729225891, −6.31551822091619936382336830855, −5.01603211109840637267955727399, −4.36016817848355593684473893836, −3.84954677915041087244712285080, −2.27640367519227128612305549158, −0.992951930922282758194273496236,
1.80484369011640605973753871355, 2.71183876263683239398026935271, 3.58084917853337387672094042373, 4.46083642926551163493894937083, 5.67663345955232880603683103412, 6.50944787024663757134735269457, 6.70874100581963838099921730131, 7.966849254152057777686276449376, 8.979239936114542836026647783256, 9.907953149108977362402598211947