L(s) = 1 | + (−1.58 − 1.58i)5-s + (−1.23 + 1.23i)7-s + 1.74i·11-s + (−0.236 − 0.236i)13-s + (4.57 + 4.57i)17-s − 6.47i·19-s + (2.82 − 2.82i)23-s + 5.00i·25-s − 0.333·29-s − 10.4·31-s + 3.90·35-s + (2.23 − 2.23i)37-s − 7.07i·41-s + (−6.47 − 6.47i)43-s + (4.57 + 4.57i)47-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)5-s + (−0.467 + 0.467i)7-s + 0.527i·11-s + (−0.0654 − 0.0654i)13-s + (1.10 + 1.10i)17-s − 1.48i·19-s + (0.589 − 0.589i)23-s + 1.00i·25-s − 0.0619·29-s − 1.88·31-s + 0.660·35-s + (0.367 − 0.367i)37-s − 1.10i·41-s + (−0.986 − 0.986i)43-s + (0.667 + 0.667i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1401472775\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1401472775\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.58 + 1.58i)T \) |
good | 7 | \( 1 + (1.23 - 1.23i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.74iT - 11T^{2} \) |
| 13 | \( 1 + (0.236 + 0.236i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.57 - 4.57i)T + 17iT^{2} \) |
| 19 | \( 1 + 6.47iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.333T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + (-2.23 + 2.23i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (6.47 + 6.47i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.57 - 4.57i)T + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 + 1.52T + 61T^{2} \) |
| 67 | \( 1 + (10.4 - 10.4i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + (9.47 + 9.47i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.52iT - 79T^{2} \) |
| 83 | \( 1 + (7.40 - 7.40i)T - 83iT^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + (-1 + i)T - 97iT^{2} \) |
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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
−8.958800657334127356286659141947, −8.543476025012071953752166005386, −7.47256760493444908189663676235, −7.08569533720481379878196929824, −5.89652215002391762839505291621, −5.27762753722915157409514721863, −4.37748975595756691248196286614, −3.60272929229370946692040772428, −2.60859350224556836005719493977, −1.33740405960395638342275940723,
0.04703360798033434470102589767, 1.48936438382254993001282648457, 3.14046378671390126894225038191, 3.33681540607101086343582225901, 4.36966885221793780263925266584, 5.45207808903430574305117929654, 6.18420517748553940915040645320, 7.12219527748298105211715331442, 7.58004232195686247859040182262, 8.251816649415892286733142076061