L(s) = 1 | + (0.800 − 0.800i)2-s + (1.09 − 1.34i)3-s + 0.718i·4-s + (−2.10 − 0.754i)5-s + (−0.199 − 1.95i)6-s + (2.17 + 2.17i)8-s + (−0.606 − 2.93i)9-s + (−2.28 + 1.08i)10-s − 5.20i·11-s + (0.964 + 0.785i)12-s + (3.24 − 3.24i)13-s + (−3.31 + 2.00i)15-s + 2.04·16-s + (−0.844 + 0.844i)17-s + (−2.83 − 1.86i)18-s + 1.32i·19-s + ⋯ |
L(s) = 1 | + (0.566 − 0.566i)2-s + (0.631 − 0.775i)3-s + 0.359i·4-s + (−0.941 − 0.337i)5-s + (−0.0813 − 0.796i)6-s + (0.769 + 0.769i)8-s + (−0.202 − 0.979i)9-s + (−0.723 + 0.341i)10-s − 1.56i·11-s + (0.278 + 0.226i)12-s + (0.900 − 0.900i)13-s + (−0.856 + 0.516i)15-s + 0.511·16-s + (−0.204 + 0.204i)17-s + (−0.668 − 0.439i)18-s + 0.302i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10308 - 1.80645i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10308 - 1.80645i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.09 + 1.34i)T \) |
| 5 | \( 1 + (2.10 + 0.754i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.800 + 0.800i)T - 2iT^{2} \) |
| 11 | \( 1 + 5.20iT - 11T^{2} \) |
| 13 | \( 1 + (-3.24 + 3.24i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.844 - 0.844i)T - 17iT^{2} \) |
| 19 | \( 1 - 1.32iT - 19T^{2} \) |
| 23 | \( 1 + (5.62 + 5.62i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 - 1.70T + 31T^{2} \) |
| 37 | \( 1 + (1.71 + 1.71i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.82iT - 41T^{2} \) |
| 43 | \( 1 + (0.281 - 0.281i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.39 + 3.39i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.51 - 3.51i)T + 53iT^{2} \) |
| 59 | \( 1 + 1.81T + 59T^{2} \) |
| 61 | \( 1 - 2.47T + 61T^{2} \) |
| 67 | \( 1 + (-7.92 - 7.92i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.06iT - 71T^{2} \) |
| 73 | \( 1 + (-1.33 + 1.33i)T - 73iT^{2} \) |
| 79 | \( 1 - 11.5iT - 79T^{2} \) |
| 83 | \( 1 + (-5.46 - 5.46i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.43T + 89T^{2} \) |
| 97 | \( 1 + (-3.06 - 3.06i)T + 97iT^{2} \) |
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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
−10.43644674190085401143260072514, −8.730640869027336475400137331725, −8.320297731572545920407073643591, −7.87885570390535191047673490957, −6.61742105227921113078437523495, −5.58292034767468335804474112118, −4.09628368411282915469488915963, −3.49156602101046821945750262783, −2.59191168567809985300304474203, −0.895137960333879079734670724126,
1.96254773071731743203194142186, 3.58741233985739886428088457082, 4.33248764762965588440538044954, 4.94726680774444363530226894382, 6.29988125498390077545829095660, 7.18304766851675133583727496057, 7.889016660788454085104896957046, 9.009312425461634259205222241400, 9.830310799505805517584073075416, 10.51668200112266393007453396356