L(s) = 1 | + (0.417 − 1.35i)2-s + (−1.65 − 1.12i)4-s − 3.04i·5-s + 7-s + (−2.21 + 1.76i)8-s + (−4.11 − 1.27i)10-s − 0.128i·11-s − 6.30i·13-s + (0.417 − 1.35i)14-s + (1.45 + 3.72i)16-s − 5.32·17-s + 6.68i·19-s + (−3.43 + 5.02i)20-s + (−0.173 − 0.0536i)22-s − 5.18·23-s + ⋯ |
L(s) = 1 | + (0.295 − 0.955i)2-s + (−0.825 − 0.563i)4-s − 1.36i·5-s + 0.377·7-s + (−0.782 + 0.622i)8-s + (−1.30 − 0.401i)10-s − 0.0387i·11-s − 1.74i·13-s + (0.111 − 0.361i)14-s + (0.364 + 0.931i)16-s − 1.29·17-s + 1.53i·19-s + (−0.767 + 1.12i)20-s + (−0.0370 − 0.0114i)22-s − 1.08·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.069472355\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.069472355\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.417 + 1.35i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + 3.04iT - 5T^{2} \) |
| 11 | \( 1 + 0.128iT - 11T^{2} \) |
| 13 | \( 1 + 6.30iT - 13T^{2} \) |
| 17 | \( 1 + 5.32T + 17T^{2} \) |
| 19 | \( 1 - 6.68iT - 19T^{2} \) |
| 23 | \( 1 + 5.18T + 23T^{2} \) |
| 29 | \( 1 + 9.96iT - 29T^{2} \) |
| 31 | \( 1 - 3.27T + 31T^{2} \) |
| 37 | \( 1 - 0.796iT - 37T^{2} \) |
| 41 | \( 1 - 2.96T + 41T^{2} \) |
| 43 | \( 1 + 6.99iT - 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 - 1.14iT - 53T^{2} \) |
| 59 | \( 1 - 11.0iT - 59T^{2} \) |
| 61 | \( 1 - 14.6iT - 61T^{2} \) |
| 67 | \( 1 - 1.05iT - 67T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 2.62T + 79T^{2} \) |
| 83 | \( 1 + 6.46iT - 83T^{2} \) |
| 89 | \( 1 - 2.23T + 89T^{2} \) |
| 97 | \( 1 + 7.88T + 97T^{2} \) |
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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
−8.937361942699386603960660714375, −8.346197987698188051740754466520, −7.78421056544743172327229548104, −5.95817015721820349518740464294, −5.58314830823384045864899681246, −4.51067721835657405783062644608, −3.99539519069217676755449087522, −2.63925566374152648972641735550, −1.53410993123720857250350194879, −0.37913651592324574856828351379,
2.13289239192001868765934972694, 3.22519883013167343594031256933, 4.31630792827219082440176395334, 4.92810242843991471588167394735, 6.34079508618146300648815699816, 6.70538445630171217391600096026, 7.23802393471703311195574951903, 8.288978849657160407286781425395, 9.090875014038096016566975999632, 9.720859777202401755703771678527