L(s) = 1 | + (0.578 + 2.15i)2-s + (1.56 − 0.737i)3-s + (−2.59 + 1.5i)4-s + (2.5 + 2.95i)6-s + (−0.684 + 2.55i)7-s + (−1.58 − 1.58i)8-s + (1.91 − 2.31i)9-s + (−2.96 + 4.26i)12-s + (−3.74 + 3.74i)13-s − 5.91·14-s + (−0.500 + 0.866i)16-s + (4.31 + 1.15i)17-s + (6.09 + 2.79i)18-s + (1.73 + i)19-s + (0.811 + 4.51i)21-s + ⋯ |
L(s) = 1 | + (0.409 + 1.52i)2-s + (0.904 − 0.425i)3-s + (−1.29 + 0.750i)4-s + (1.02 + 1.20i)6-s + (−0.258 + 0.965i)7-s + (−0.559 − 0.559i)8-s + (0.637 − 0.770i)9-s + (−0.856 + 1.23i)12-s + (−1.03 + 1.03i)13-s − 1.58·14-s + (−0.125 + 0.216i)16-s + (1.04 + 0.280i)17-s + (1.43 + 0.658i)18-s + (0.397 + 0.229i)19-s + (0.177 + 0.984i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.894325 + 2.01420i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.894325 + 2.01420i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.56 + 0.737i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.684 - 2.55i)T \) |
good | 2 | \( 1 + (-0.578 - 2.15i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.74 - 3.74i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.31 - 1.15i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.73 - i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.47 + 1.73i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-10.2 + 2.73i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 5.91iT - 41T^{2} \) |
| 43 | \( 1 + (1.87 - 1.87i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.31 + 8.63i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.15 - 4.31i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (5.91 + 10.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.684 + 2.55i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 + (-5.11 - 1.36i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.73 - i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.58 + 1.58i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.95 + 5.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.48 + 7.48i)T + 97iT^{2} \) |
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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
−11.41731671657034725183554165991, −9.683590808768060452132953369610, −9.168730575272928548869686277327, −8.251821968278783912369495514854, −7.45916608459692923977394671342, −6.77383385414665217479868240265, −5.80427578529506890614187294799, −4.83850493247961145681195465730, −3.56216895646501509585567267820, −2.17111134602611000319221239002,
1.15755483147547056983736325371, 2.80563698804356765437694665974, 3.31196593510279501479496793349, 4.42967651583522880914147713604, 5.26375273837005429513943977421, 7.24363572822499486045152820239, 7.85806317477038940204470540683, 9.404317761861552623422421542607, 9.742506700960741538899930776745, 10.54924709861756370128040274521