| L(s) = 1 | + (−0.465 − 0.806i)2-s + (−2.42 − 1.75i)3-s + (0.566 − 0.980i)4-s + (1.07 − 1.19i)5-s + (−0.291 + 2.77i)6-s + (−0.384 + 0.427i)7-s − 2.91·8-s + (1.84 + 5.66i)9-s + (−1.46 − 0.312i)10-s + (−1.96 − 0.873i)11-s + (−3.09 + 1.37i)12-s + (−0.715 + 0.318i)13-s + (0.524 + 0.111i)14-s + (−4.71 + 1.00i)15-s + (0.226 + 0.392i)16-s + (2.30 − 0.489i)17-s + ⋯ |
| L(s) = 1 | + (−0.329 − 0.570i)2-s + (−1.39 − 1.01i)3-s + (0.283 − 0.490i)4-s + (0.482 − 0.535i)5-s + (−0.118 + 1.13i)6-s + (−0.145 + 0.161i)7-s − 1.03·8-s + (0.613 + 1.88i)9-s + (−0.464 − 0.0987i)10-s + (−0.591 − 0.263i)11-s + (−0.893 + 0.397i)12-s + (−0.198 + 0.0883i)13-s + (0.140 + 0.0297i)14-s + (−1.21 + 0.258i)15-s + (0.0566 + 0.0981i)16-s + (0.559 − 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0100855 + 0.571888i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0100855 + 0.571888i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-8.84 - 8.53i)T \) |
| good | 2 | \( 1 + (0.465 + 0.806i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (2.42 + 1.75i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (-1.07 + 1.19i)T + (-0.522 - 4.97i)T^{2} \) |
| 7 | \( 1 + (0.384 - 0.427i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (1.96 + 0.873i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (0.715 - 0.318i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-2.30 + 0.489i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 23 | \( 1 + (3.95 + 6.84i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.76 + 2.73i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-8.86 + 1.88i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-0.862 + 8.20i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (9.49 + 6.89i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (1.99 + 2.21i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (8.66 - 3.85i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-10.5 + 7.64i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 - 6.99T + 59T^{2} \) |
| 61 | \( 1 + (-0.271 - 2.58i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-1.87 + 5.77i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.47 - 0.525i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-4.29 - 13.2i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.259 + 0.799i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (2.36 - 7.27i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.285 - 0.316i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-10.5 + 2.23i)T + (88.6 - 39.4i)T^{2} \) |
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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
−12.22460616363335515342579678127, −11.65515787209974870258762301282, −10.57042144359791350341526533334, −9.847222019007283568436076254720, −8.315975941285596126260361925645, −6.83872713115857951461243490622, −5.92007502681304469917703837165, −5.14187354305866325635221383482, −2.24442670393881963928012411476, −0.72663299086774824016160168540,
3.27748178094153169510039504656, 4.93935749953210838091568702319, 6.05859971727938488943232502760, 6.86215376108109898057440557692, 8.225114207762833876595072509046, 9.962070837469333679385384708125, 10.13784161910846986632306428451, 11.59029724338070171312104251249, 12.05364878512619119829362838529, 13.48084313453601566203118586694
