L(s) = 1 | + (−0.895 + 1.55i)3-s − 3.79·5-s + (−0.5 − 0.866i)7-s + (−0.104 − 0.180i)9-s + (1.89 − 3.28i)11-s + (2.5 − 2.59i)13-s + (3.39 − 5.88i)15-s + (−1.5 − 2.59i)17-s + (−3.68 − 6.38i)19-s + 1.79·21-s + (−3 + 5.19i)23-s + 9.37·25-s − 5·27-s + (1.10 − 1.91i)29-s − 31-s + ⋯ |
L(s) = 1 | + (−0.517 + 0.895i)3-s − 1.69·5-s + (−0.188 − 0.327i)7-s + (−0.0347 − 0.0602i)9-s + (0.571 − 0.989i)11-s + (0.693 − 0.720i)13-s + (0.876 − 1.51i)15-s + (−0.363 − 0.630i)17-s + (−0.845 − 1.46i)19-s + 0.390·21-s + (−0.625 + 1.08i)23-s + 1.87·25-s − 0.962·27-s + (0.205 − 0.355i)29-s − 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.304122 - 0.300246i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.304122 - 0.300246i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (-2.5 + 2.59i)T \) |
good | 3 | \( 1 + (0.895 - 1.55i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 11 | \( 1 + (-1.89 + 3.28i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.68 + 6.38i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.10 + 1.91i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.791 + 1.37i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.18 + 3.78i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.79 + 6.56i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 + (-2.68 + 4.65i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.18 - 15.9i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.25690332274593772350584002423, −10.62150186122591898182108291972, −9.357618311228356701398399246132, −8.389949470308854772880028733866, −7.52071530922316272284849211141, −6.36788658552930167537781036919, −5.02040077863399731901928291529, −4.08097707525387481797908406776, −3.33785266248429629243502448031, −0.31962908619490817082430572195,
1.65732336397060433724793382099, 3.72126341657009647447545791621, 4.47224557371159307375186829560, 6.32354088944258437282584434203, 6.74339628268830533111895161162, 7.921293429880098905389763300372, 8.516076255122149646769555655149, 9.875970960468925520630866234117, 11.14519595795780124538184014377, 11.75478887042410449066470024488