L(s) = 1 | + (1.83 − 1.28i)5-s + (2.94 − 2.94i)7-s − 1.61·11-s + (2.50 + 2.50i)13-s + (4.59 + 4.59i)17-s − 4i·19-s + (−1.09 − 1.09i)23-s + (1.70 − 4.70i)25-s + 4.75·29-s + 5.01i·31-s + (1.61 − 9.18i)35-s + (−2.50 + 2.50i)37-s − 9.18·41-s + (7.40 − 7.40i)43-s + (−7.32 + 7.32i)47-s + ⋯ |
L(s) = 1 | + (0.818 − 0.574i)5-s + (1.11 − 1.11i)7-s − 0.485·11-s + (0.696 + 0.696i)13-s + (1.11 + 1.11i)17-s − 0.917i·19-s + (−0.227 − 0.227i)23-s + (0.340 − 0.940i)25-s + 0.882·29-s + 0.901i·31-s + (0.272 − 1.55i)35-s + (−0.412 + 0.412i)37-s − 1.43·41-s + (1.12 − 1.12i)43-s + (−1.06 + 1.06i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.305117060\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.305117060\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.83 + 1.28i)T \) |
good | 7 | \( 1 + (-2.94 + 2.94i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + (-2.50 - 2.50i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.59 - 4.59i)T + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (1.09 + 1.09i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 - 5.01iT - 31T^{2} \) |
| 37 | \( 1 + (2.50 - 2.50i)T - 37iT^{2} \) |
| 41 | \( 1 + 9.18T + 41T^{2} \) |
| 43 | \( 1 + (-7.40 + 7.40i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.32 - 7.32i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.11 + 3.11i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.61iT - 59T^{2} \) |
| 61 | \( 1 + 6.78iT - 61T^{2} \) |
| 67 | \( 1 + (7.40 + 7.40i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-5 + 5i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.01T + 79T^{2} \) |
| 83 | \( 1 + (7.57 - 7.57i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.74iT - 89T^{2} \) |
| 97 | \( 1 + (-2.40 - 2.40i)T + 97iT^{2} \) |
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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
−9.432558005551660164208100991052, −8.473601848903127621279995747423, −8.043850030055525459055109299038, −6.96781824754719608669688480327, −6.15327791469507990523435561779, −5.08347370401666617121518206638, −4.55084606834505810498723958471, −3.43759465624749890345680072203, −1.88662775181260967165803767225, −1.09653385534773946600416758701,
1.43467189775693613347603248088, 2.48743410681164082232955616355, 3.33226387370333586954628408890, 4.87201668270266440582496373346, 5.59572618470658827603183472822, 6.05027262877214899274285661444, 7.33824576464227006046008300506, 8.074995958307006329270705614344, 8.740321623017161305229759718983, 9.775488793304138246167628902223