L(s) = 1 | + (0.0922 + 0.995i)3-s + (0.932 + 0.361i)4-s + (0.397 + 0.798i)7-s + (−0.982 + 0.183i)9-s + (−0.273 + 0.961i)12-s + (−0.404 − 0.368i)13-s + (0.739 + 0.673i)16-s + (−1.25 − 1.14i)19-s + (−0.757 + 0.469i)21-s + (0.739 − 0.673i)25-s + (−0.273 − 0.961i)27-s + (0.0822 + 0.887i)28-s + (0.0822 − 0.887i)31-s + (−0.982 − 0.183i)36-s + (0.831 + 1.66i)37-s + ⋯ |
L(s) = 1 | + (0.0922 + 0.995i)3-s + (0.932 + 0.361i)4-s + (0.397 + 0.798i)7-s + (−0.982 + 0.183i)9-s + (−0.273 + 0.961i)12-s + (−0.404 − 0.368i)13-s + (0.739 + 0.673i)16-s + (−1.25 − 1.14i)19-s + (−0.757 + 0.469i)21-s + (0.739 − 0.673i)25-s + (−0.273 − 0.961i)27-s + (0.0822 + 0.887i)28-s + (0.0822 − 0.887i)31-s + (−0.982 − 0.183i)36-s + (0.831 + 1.66i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.235091026\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.235091026\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0922 - 0.995i)T \) |
| 307 | \( 1 + (-0.0922 - 0.995i)T \) |
good | 2 | \( 1 + (-0.932 - 0.361i)T^{2} \) |
| 5 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 7 | \( 1 + (-0.397 - 0.798i)T + (-0.602 + 0.798i)T^{2} \) |
| 11 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 13 | \( 1 + (0.404 + 0.368i)T + (0.0922 + 0.995i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (1.25 + 1.14i)T + (0.0922 + 0.995i)T^{2} \) |
| 23 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 29 | \( 1 + (-0.932 - 0.361i)T^{2} \) |
| 31 | \( 1 + (-0.0822 + 0.887i)T + (-0.982 - 0.183i)T^{2} \) |
| 37 | \( 1 + (-0.831 - 1.66i)T + (-0.602 + 0.798i)T^{2} \) |
| 41 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 43 | \( 1 + (1.58 + 0.981i)T + (0.445 + 0.895i)T^{2} \) |
| 47 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 61 | \( 1 + (-0.465 - 0.288i)T + (0.445 + 0.895i)T^{2} \) |
| 67 | \( 1 + (0.404 - 1.42i)T + (-0.850 - 0.526i)T^{2} \) |
| 71 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 73 | \( 1 + (-0.658 - 0.600i)T + (0.0922 + 0.995i)T^{2} \) |
| 79 | \( 1 + (0.156 - 0.0971i)T + (0.445 - 0.895i)T^{2} \) |
| 83 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 89 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 97 | \( 1 + (0.404 - 0.368i)T + (0.0922 - 0.995i)T^{2} \) |
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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
−10.48193138806353299220862442005, −9.777085656078424114190013807115, −8.556544564447230151333051455934, −8.327824183586344374694742554311, −7.00748686087160736688998796048, −6.13126534208459290446862788177, −5.16027179520275002412170722927, −4.25990634018578108320860716651, −2.95461774199377805378793425274, −2.29165457711634037910696193500,
1.35833926087354270853112896696, 2.27404520036145013921286146386, 3.54967856420566166929930059918, 4.95272707704674969589009802115, 6.08695068245160772776051916220, 6.72626553826092588548374366915, 7.49717575249396683658626194678, 8.112073014624085454674608943259, 9.209189950419170844598470843878, 10.37832248646549592366690838498