| L(s) = 1 | + (−0.891 − 0.453i)2-s + (−1.64 − 0.530i)3-s + (0.587 + 0.809i)4-s + (1.35 + 1.78i)5-s + (1.22 + 1.22i)6-s + (−0.152 − 0.152i)7-s + (−0.156 − 0.987i)8-s + (2.43 + 1.75i)9-s + (−0.396 − 2.20i)10-s + (4.88 − 1.58i)11-s + (−0.539 − 1.64i)12-s + (0.674 + 1.32i)13-s + (0.0667 + 0.205i)14-s + (−1.28 − 3.65i)15-s + (−0.309 + 0.951i)16-s + (4.81 − 0.762i)17-s + ⋯ |
| L(s) = 1 | + (−0.630 − 0.321i)2-s + (−0.951 − 0.306i)3-s + (0.293 + 0.404i)4-s + (0.604 + 0.796i)5-s + (0.501 + 0.498i)6-s + (−0.0577 − 0.0577i)7-s + (−0.0553 − 0.349i)8-s + (0.812 + 0.583i)9-s + (−0.125 − 0.695i)10-s + (1.47 − 0.478i)11-s + (−0.155 − 0.475i)12-s + (0.187 + 0.367i)13-s + (0.0178 + 0.0548i)14-s + (−0.331 − 0.943i)15-s + (−0.0772 + 0.237i)16-s + (1.16 − 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.748942 - 0.0183233i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.748942 - 0.0183233i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.891 + 0.453i)T \) |
| 3 | \( 1 + (1.64 + 0.530i)T \) |
| 5 | \( 1 + (-1.35 - 1.78i)T \) |
| good | 7 | \( 1 + (0.152 + 0.152i)T + 7iT^{2} \) |
| 11 | \( 1 + (-4.88 + 1.58i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.674 - 1.32i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-4.81 + 0.762i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.283 + 0.389i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.21 - 2.38i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (7.59 - 5.51i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.84 + 1.33i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.83 + 1.95i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-5.95 - 1.93i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (2.72 - 2.72i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.58 + 10.0i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (7.59 + 1.20i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (1.54 - 4.74i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (4.21 + 12.9i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-2.35 - 14.8i)T + (-63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (7.13 + 9.81i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-8.36 - 4.26i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (1.28 + 1.76i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (0.782 + 4.93i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (3.38 + 10.4i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (9.96 + 1.57i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.78766154646046975953428347274, −11.62843825374250316627570677498, −11.15436234166460439731515499478, −10.02139394854013351872436578866, −9.206505445962535219158805293277, −7.53386157268329790925022338133, −6.61942879068706168964365603441, −5.66246711789462304290592773618, −3.62700334965096990048523178296, −1.57186513919337259224616282515,
1.30718250024942531144448097366, 4.22543265728786842936393438577, 5.61872597763498726167281188045, 6.35918011203026235600006177876, 7.77033663914553789292676719920, 9.275019736479074816221269614009, 9.707400353462930945531221686192, 10.88992177768555908954746458664, 12.02272935948653835537412861189, 12.70179852296645268050115234252
