L(s) = 1 | + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s + (1.73 + i)13-s + (−0.499 − 0.866i)16-s + 19-s + 0.999i·28-s + (0.5 − 0.866i)31-s − i·37-s + (0.866 − 0.5i)43-s + (1.73 − 0.999i)52-s + (0.5 + 0.866i)61-s − 0.999·64-s + (−1.73 − i)67-s + i·73-s + (0.5 − 0.866i)76-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s + (1.73 + i)13-s + (−0.499 − 0.866i)16-s + 19-s + 0.999i·28-s + (0.5 − 0.866i)31-s − i·37-s + (0.866 − 0.5i)43-s + (1.73 − 0.999i)52-s + (0.5 + 0.866i)61-s − 0.999·64-s + (−1.73 − i)67-s + i·73-s + (0.5 − 0.866i)76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.298321952\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.298321952\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
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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
−9.263363718829593731497623773546, −8.798891862954446786313430683094, −7.58496833675444800750523992766, −6.74967848808302019581729487642, −6.04055870999586731794032196622, −5.64483421598684551117399624383, −4.37237449095198621933523325672, −3.38699022628688049771200366029, −2.33384816977556655199343047346, −1.18813921089393403007175366882,
1.27111970955611383956660261040, 2.96159028218511197966092612328, 3.34907027759168016892439827197, 4.24930505098907511555571760822, 5.57077216309624019353381112317, 6.37769374262913389587980367609, 7.00045604406334869669917411262, 7.898740605560842738238515172294, 8.439125676540347282566296845211, 9.318811398668117819633076283477