L(s) = 1 | + 0.917i·2-s + (−0.321 − 1.70i)3-s + 1.15·4-s + (0.5 − 0.866i)5-s + (1.56 − 0.295i)6-s + (0.697 + 2.55i)7-s + 2.89i·8-s + (−2.79 + 1.09i)9-s + (0.794 + 0.458i)10-s + (2.88 − 1.66i)11-s + (−0.372 − 1.96i)12-s + (2.08 − 1.20i)13-s + (−2.34 + 0.639i)14-s + (−1.63 − 0.572i)15-s − 0.345·16-s + (0.514 − 0.890i)17-s + ⋯ |
L(s) = 1 | + 0.649i·2-s + (−0.185 − 0.982i)3-s + 0.578·4-s + (0.223 − 0.387i)5-s + (0.637 − 0.120i)6-s + (0.263 + 0.964i)7-s + 1.02i·8-s + (−0.930 + 0.365i)9-s + (0.251 + 0.145i)10-s + (0.869 − 0.501i)11-s + (−0.107 − 0.568i)12-s + (0.578 − 0.333i)13-s + (−0.626 + 0.171i)14-s + (−0.422 − 0.147i)15-s − 0.0863·16-s + (0.124 − 0.216i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.57349 + 0.0544219i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57349 + 0.0544219i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.321 + 1.70i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.697 - 2.55i)T \) |
good | 2 | \( 1 - 0.917iT - 2T^{2} \) |
| 11 | \( 1 + (-2.88 + 1.66i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.08 + 1.20i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.514 + 0.890i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.75 + 2.74i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.46 + 3.73i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.78 + 2.18i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.525iT - 31T^{2} \) |
| 37 | \( 1 + (-3.83 - 6.64i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0682 - 0.118i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.65 - 9.79i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.02T + 47T^{2} \) |
| 53 | \( 1 + (1.24 + 0.716i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 0.639T + 59T^{2} \) |
| 61 | \( 1 + 10.2iT - 61T^{2} \) |
| 67 | \( 1 + 4.05T + 67T^{2} \) |
| 71 | \( 1 - 4.03iT - 71T^{2} \) |
| 73 | \( 1 + (-5.39 - 3.11i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 2.55T + 79T^{2} \) |
| 83 | \( 1 + (-2.30 + 3.99i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.33 - 12.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.8 + 8.58i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.62197268112598206410550412152, −11.25830711476888042468386551070, −9.561922367081337238932557125051, −8.395343846035666005528978739828, −7.941976369226017284423809330378, −6.57996352731789856525293168434, −6.04430317624027584872485572636, −5.13016436667892352395287658379, −2.90990937057509411539113539543, −1.56629122672643489971897668607,
1.65813655106231817913338201511, 3.49456892923366831231214237654, 4.04450009196311766391802838222, 5.68088215297344206111342041446, 6.73089424088533973358206246091, 7.77026098710851939582933958304, 9.346120222494711191760477129563, 10.00378037079795398299533591277, 10.71029954254102162546496724085, 11.51484099819606940278161586610