L(s) = 1 | + (0.581 + 1.28i)2-s + (−0.707 − 0.707i)3-s + (−1.32 + 1.50i)4-s + (0.500 − 1.32i)6-s + (−1.41 + 1.41i)7-s + (−2.70 − 0.832i)8-s + 1.00i·9-s + 5.29i·11-s + (1.99 − 0.125i)12-s + (−3.74 + 3.74i)13-s + (−2.64 − 1.00i)14-s + (−0.5 − 3.96i)16-s + (−1.28 + 0.581i)18-s + 5.29·19-s + 2.00·21-s + (−6.82 + 3.07i)22-s + ⋯ |
L(s) = 1 | + (0.411 + 0.911i)2-s + (−0.408 − 0.408i)3-s + (−0.661 + 0.750i)4-s + (0.204 − 0.540i)6-s + (−0.534 + 0.534i)7-s + (−0.955 − 0.294i)8-s + 0.333i·9-s + 1.59i·11-s + (0.576 − 0.0361i)12-s + (−1.03 + 1.03i)13-s + (−0.707 − 0.267i)14-s + (−0.125 − 0.992i)16-s + (−0.303 + 0.137i)18-s + 1.21·19-s + 0.436·21-s + (−1.45 + 0.656i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.228043 + 0.910331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.228043 + 0.910331i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.581 - 1.28i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.41 - 1.41i)T - 7iT^{2} \) |
| 11 | \( 1 - 5.29iT - 11T^{2} \) |
| 13 | \( 1 + (3.74 - 3.74i)T - 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 - 5.29iT - 31T^{2} \) |
| 37 | \( 1 + (-3.74 - 3.74i)T + 37iT^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + (-5.65 - 5.65i)T + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-7.48 + 7.48i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + (-8.48 + 8.48i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (7.48 - 7.48i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + (-7.48 - 7.48i)T + 97iT^{2} \) |
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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
−12.19225626828744473095159948614, −11.77276670574379259634913751633, −9.872743707304082871620040065404, −9.390407723287935901078127303132, −7.962944173410294225539948901509, −7.10318784973835083574842739297, −6.39614891026667465664209358085, −5.19113952404350885894387436936, −4.29864171522189442654944847211, −2.46974784227322696232601674249,
0.63511398975853039823259255291, 2.96486803733508353378127532663, 3.81327161103452755568936856090, 5.27238388607455772792031778286, 5.90512490069885545510684548930, 7.45971766363412715861701030001, 8.853753657203960796457193039296, 9.827862752689409637193911280504, 10.49608470973322400812445688612, 11.32133714762949500202879265536