L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)6-s − 0.329·7-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−1.55 + 4.78i)11-s + (0.309 + 0.951i)12-s + (−0.148 − 0.458i)13-s + (−0.101 + 0.313i)14-s + (0.309 + 0.951i)16-s + (−5.49 + 3.98i)17-s + 18-s + (−4.40 + 3.19i)19-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.330 + 0.239i)6-s − 0.124·7-s + (−0.286 + 0.207i)8-s + (0.103 + 0.317i)9-s + (−0.468 + 1.44i)11-s + (0.0892 + 0.274i)12-s + (−0.0413 − 0.127i)13-s + (−0.0271 + 0.0837i)14-s + (0.0772 + 0.237i)16-s + (−1.33 + 0.967i)17-s + 0.235·18-s + (−1.00 + 0.733i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.540758 + 0.340495i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.540758 + 0.340495i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.329T + 7T^{2} \) |
| 11 | \( 1 + (1.55 - 4.78i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.148 + 0.458i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (5.49 - 3.98i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (4.40 - 3.19i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.00 + 6.18i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.87 - 3.53i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.06 - 0.775i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.241 - 0.741i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.86 - 11.9i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 + (-3.54 - 2.57i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.36 - 0.990i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.313 - 0.966i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.29 + 3.98i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (2.54 - 1.84i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (4.62 + 3.35i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.909 + 2.79i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (6.86 + 4.98i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (13.9 - 10.1i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.06 + 3.28i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-7.69 - 5.58i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.53393269006657033662019176226, −10.01655259880259985178616709415, −8.843077173112177504915971122993, −8.018529431078495286164502516898, −6.80684879331855663881159492075, −6.16836219267839531753838807325, −4.80430779792861705392841547473, −4.29354554232069085308915737549, −2.67107963208851466272761067391, −1.66864292387223894586382988635,
0.31285239038542887254119292087, 2.69319120662854101075978637911, 3.92870600118873390211056345863, 4.92052970467497234054536367023, 5.75634335992465227185487688802, 6.58770605362732776688092664668, 7.42562515648998665672230152867, 8.644277195209368627692009553765, 9.074551991413221428646534771412, 10.23644045737676754909477240678