L(s) = 1 | + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s − 1.41i·5-s + 7-s − 2.82i·8-s + (−1.00 − 1.73i)10-s − 1.41i·11-s + (1.22 − 0.707i)14-s + (−2.00 − 3.46i)16-s − 2.44·17-s + 3.46i·19-s + (−2.44 − 1.41i)20-s + (−1.00 − 1.73i)22-s + 2.44·23-s + 2.99·25-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)2-s + (0.499 − 0.866i)4-s − 0.632i·5-s + 0.377·7-s − 0.999i·8-s + (−0.316 − 0.547i)10-s − 0.426i·11-s + (0.327 − 0.188i)14-s + (−0.500 − 0.866i)16-s − 0.594·17-s + 0.794i·19-s + (−0.547 − 0.316i)20-s + (−0.213 − 0.369i)22-s + 0.510·23-s + 0.599·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67060 - 1.67060i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67060 - 1.67060i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 3.46iT - 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 7.34T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.88491248071221724939226717262, −10.02156961004943414571443464077, −9.006559617978721619799465921491, −8.050535525008617212301035740878, −6.78347985266239099448637801591, −5.79496995979326825796546263568, −4.87223307419506817932320292005, −3.99966843890866509262109421250, −2.66300379041176097578364095511, −1.20982805244815938810505320924,
2.23893208315553379262902045330, 3.38340487141076566696957292529, 4.59216245939074463375673807545, 5.41732281679631057215162647542, 6.75669555127918664552439288560, 7.09702954877581353712381914170, 8.287355640742835929465707995130, 9.180765534108394816574388074260, 10.64408566649989664700194609127, 11.12566194720480836714014333715