L(s) = 1 | + (1.22 + 0.707i)5-s + (−1.22 + 0.707i)11-s + (−0.5 + 0.866i)13-s + 1.41i·17-s − 19-s + (1.22 + 0.707i)23-s + (0.499 + 0.866i)25-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)43-s + (0.5 − 0.866i)49-s − 1.41i·53-s − 2·55-s + (1.22 + 0.707i)59-s + (−0.5 − 0.866i)61-s + (−1.22 + 0.707i)65-s + ⋯ |
L(s) = 1 | + (1.22 + 0.707i)5-s + (−1.22 + 0.707i)11-s + (−0.5 + 0.866i)13-s + 1.41i·17-s − 19-s + (1.22 + 0.707i)23-s + (0.499 + 0.866i)25-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)43-s + (0.5 − 0.866i)49-s − 1.41i·53-s − 2·55-s + (1.22 + 0.707i)59-s + (−0.5 − 0.866i)61-s + (−1.22 + 0.707i)65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.252330405\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.252330405\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.713630440254585182685485664068, −8.834811638999707460153843334134, −7.948739155887996312690728228452, −6.99659644414286080535524041167, −6.45466609130543389223729429392, −5.56145697471835527908926815234, −4.81183598072536606319410544602, −3.66668028253548222431854159140, −2.36459670614903650188329289796, −1.95627933448462858797820254470,
0.901422944492881128130064384825, 2.43922234660793094903985493600, 2.98412080865379649076775476307, 4.70709254021752321848737017734, 5.16963566534912335334275710656, 5.87238794948940381037609796957, 6.79927045563676583991826741458, 7.76724906788175358475625353278, 8.577977661282312411864948490155, 9.178768292855885242936909464845