| L(s) = 1 | + (1.17 + 0.779i)2-s + (1.19 − 1.25i)3-s + (0.784 + 1.83i)4-s + (0.0402 − 2.23i)5-s + (2.38 − 0.543i)6-s − 1.15·7-s + (−0.509 + 2.78i)8-s + (−0.134 − 2.99i)9-s + (1.79 − 2.60i)10-s + (0.602 + 1.85i)11-s + (3.24 + 1.22i)12-s + (6.39 + 2.07i)13-s + (−1.36 − 0.898i)14-s + (−2.75 − 2.72i)15-s + (−2.76 + 2.88i)16-s + (−4.07 − 2.95i)17-s + ⋯ |
| L(s) = 1 | + (0.834 + 0.551i)2-s + (0.691 − 0.722i)3-s + (0.392 + 0.919i)4-s + (0.0179 − 0.999i)5-s + (0.975 − 0.222i)6-s − 0.435·7-s + (−0.179 + 0.983i)8-s + (−0.0447 − 0.998i)9-s + (0.566 − 0.824i)10-s + (0.181 + 0.559i)11-s + (0.935 + 0.352i)12-s + (1.77 + 0.575i)13-s + (−0.363 − 0.240i)14-s + (−0.710 − 0.703i)15-s + (−0.692 + 0.721i)16-s + (−0.987 − 0.717i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.40334 - 0.0228481i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.40334 - 0.0228481i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.17 - 0.779i)T \) |
| 3 | \( 1 + (-1.19 + 1.25i)T \) |
| 5 | \( 1 + (-0.0402 + 2.23i)T \) |
| good | 7 | \( 1 + 1.15T + 7T^{2} \) |
| 11 | \( 1 + (-0.602 - 1.85i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-6.39 - 2.07i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (4.07 + 2.95i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.112 + 0.155i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (5.77 - 1.87i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.480 - 0.661i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (3.70 - 5.10i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.47 - 0.480i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-6.34 - 2.06i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 4.89T + 43T^{2} \) |
| 47 | \( 1 + (-2.70 - 3.72i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.10 + 2.98i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.85 - 8.80i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (1.91 + 5.88i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (7.13 + 5.18i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-10.7 + 7.80i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.62 + 1.50i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-6.10 - 8.39i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-7.95 + 10.9i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (1.79 - 0.583i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (5.18 + 7.13i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−12.14100074914655311675679608903, −11.24343947077638318094840602874, −9.373277599036469156493105067736, −8.730651416383993448423970483006, −7.83781196809983907015626900119, −6.72783777506003304762790410723, −5.96839501823982727356605206710, −4.48839780010613380638735519725, −3.50741189348113458183533764354, −1.84151319320130544346019272683,
2.26600303995188692842663678077, 3.50797171684180098565182907518, 4.01535790518923226802636362603, 5.78779326684695544311692722026, 6.46111127135453840919482446121, 8.017564981599037045608742600558, 9.130612306646739940384596693148, 10.22336062006368784298231212588, 10.81940068088660420755429522269, 11.44314793820254490542924903422
