L(s) = 1 | − 2.82·5-s + (1.73 + 2i)7-s + 4.89·11-s + 6·13-s + 4.89i·17-s − 8.48i·23-s + 3.00·25-s + 3.46·31-s + (−4.89 − 5.65i)35-s − 6.92i·37-s + 4.89i·41-s + 3.46·43-s − 9.79·47-s + (−1.00 + 6.92i)49-s − 9.79i·53-s + ⋯ |
L(s) = 1 | − 1.26·5-s + (0.654 + 0.755i)7-s + 1.47·11-s + 1.66·13-s + 1.18i·17-s − 1.76i·23-s + 0.600·25-s + 0.622·31-s + (−0.828 − 0.956i)35-s − 1.13i·37-s + 0.765i·41-s + 0.528·43-s − 1.42·47-s + (−0.142 + 0.989i)49-s − 1.34i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.960601503\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.960601503\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 4.89iT - 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + 9.79iT - 53T^{2} \) |
| 59 | \( 1 - 5.65iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 5.65iT - 83T^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−8.409664255378804194848969647556, −8.148267520113372026129049541823, −6.92151456466506536253213646152, −6.35122768172717986066038954866, −5.64516669613222469702800910403, −4.35101310397607466443180067991, −4.08329015168181078216153306684, −3.22852104590646133007663059317, −1.91571470477267452135372930172, −0.921576956845613378767616426951,
0.827658362095883968433396681585, 1.55713160420239701321147716183, 3.30922860052487716512540987952, 3.76694195084383060561113508660, 4.39814554843849250266046777002, 5.28594014307674244328177339453, 6.40793867620285264904382613332, 6.95382367003710302094031943047, 7.79617666609163259047837943755, 8.182513156264960756383676588198