L(s) = 1 | + (−0.0564 − 1.41i)2-s + (−2.95 − 1.22i)3-s + (−1.99 + 0.159i)4-s + (−1.21 − 2.92i)5-s + (−1.56 + 4.24i)6-s + (0.707 − 0.707i)7-s + (0.338 + 2.80i)8-s + (5.11 + 5.11i)9-s + (−4.06 + 1.87i)10-s + (−1.89 + 0.786i)11-s + (6.08 + 1.96i)12-s + (1.39 − 3.36i)13-s + (−1.03 − 0.959i)14-s + 10.1i·15-s + (3.94 − 0.636i)16-s + 1.19i·17-s + ⋯ |
L(s) = 1 | + (−0.0399 − 0.999i)2-s + (−1.70 − 0.706i)3-s + (−0.996 + 0.0797i)4-s + (−0.541 − 1.30i)5-s + (−0.638 + 1.73i)6-s + (0.267 − 0.267i)7-s + (0.119 + 0.992i)8-s + (1.70 + 1.70i)9-s + (−1.28 + 0.593i)10-s + (−0.572 + 0.237i)11-s + (1.75 + 0.568i)12-s + (0.386 − 0.932i)13-s + (−0.277 − 0.256i)14-s + 2.61i·15-s + (0.987 − 0.159i)16-s + 0.289i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00334 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00334 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.166911 + 0.167470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.166911 + 0.167470i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0564 + 1.41i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (2.95 + 1.22i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (1.21 + 2.92i)T + (-3.53 + 3.53i)T^{2} \) |
| 11 | \( 1 + (1.89 - 0.786i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-1.39 + 3.36i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 1.19iT - 17T^{2} \) |
| 19 | \( 1 + (1.98 - 4.80i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (4.77 + 4.77i)T + 23iT^{2} \) |
| 29 | \( 1 + (1.61 + 0.670i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 1.81T + 31T^{2} \) |
| 37 | \( 1 + (-3.48 - 8.41i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (6.87 + 6.87i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.37 + 0.984i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 1.33iT - 47T^{2} \) |
| 53 | \( 1 + (5.26 - 2.17i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-3.13 - 7.57i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (4.76 + 1.97i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (9.61 + 3.98i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-6.26 + 6.26i)T - 71iT^{2} \) |
| 73 | \( 1 + (-0.436 - 0.436i)T + 73iT^{2} \) |
| 79 | \( 1 - 4.79iT - 79T^{2} \) |
| 83 | \( 1 + (-2.57 + 6.22i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (8.88 - 8.88i)T - 89iT^{2} \) |
| 97 | \( 1 + 5.09T + 97T^{2} \) |
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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
−11.74902838191480490163512161020, −10.71314662401788778612412831084, −10.14639647220807290122284239646, −8.402477866696972242452328008776, −7.76079521395747544583893267796, −5.97206552054773086575898280152, −5.07565746502223809530356177244, −4.19858496082960933180447825556, −1.56928604677385177318322933142, −0.25850699132329544709744865360,
3.77877944309056878755989657876, 4.85731637969166190703103483259, 5.94233575517654275220500194225, 6.69377025115830688279416763038, 7.59302362416472301093065939176, 9.199995390602436559286392302090, 10.21624168022044654539102047777, 11.15281485340287628650177520869, 11.59441647375925432814992600496, 12.92346490665306698374637563978