L(s) = 1 | + 2-s + (−0.571 − 1.63i)3-s + 4-s + (−2.12 + 0.689i)5-s + (−0.571 − 1.63i)6-s − 1.28·7-s + 8-s + (−2.34 + 1.86i)9-s + (−2.12 + 0.689i)10-s − 4.52·11-s + (−0.571 − 1.63i)12-s + 3.04i·13-s − 1.28·14-s + (2.34 + 3.08i)15-s + 16-s + 5.73i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.330 − 0.943i)3-s + 0.5·4-s + (−0.951 + 0.308i)5-s + (−0.233 − 0.667i)6-s − 0.487·7-s + 0.353·8-s + (−0.781 + 0.623i)9-s + (−0.672 + 0.217i)10-s − 1.36·11-s + (−0.165 − 0.471i)12-s + 0.843i·13-s − 0.344·14-s + (0.605 + 0.796i)15-s + 0.250·16-s + 1.39i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.126960 + 0.265045i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.126960 + 0.265045i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.571 + 1.63i)T \) |
| 5 | \( 1 + (2.12 - 0.689i)T \) |
| 23 | \( 1 + (4.79 - 0.132i)T \) |
good | 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 - 3.04iT - 13T^{2} \) |
| 17 | \( 1 - 5.73iT - 17T^{2} \) |
| 19 | \( 1 + 3.52iT - 19T^{2} \) |
| 29 | \( 1 - 3.03iT - 29T^{2} \) |
| 31 | \( 1 + 7.30T + 31T^{2} \) |
| 37 | \( 1 - 8.69T + 37T^{2} \) |
| 41 | \( 1 + 12.7iT - 41T^{2} \) |
| 43 | \( 1 + 8.85T + 43T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 - 4.54iT - 53T^{2} \) |
| 59 | \( 1 - 12.7iT - 59T^{2} \) |
| 61 | \( 1 + 3.04iT - 61T^{2} \) |
| 67 | \( 1 + 3.98T + 67T^{2} \) |
| 71 | \( 1 + 5.30iT - 71T^{2} \) |
| 73 | \( 1 - 9.43iT - 73T^{2} \) |
| 79 | \( 1 + 16.0iT - 79T^{2} \) |
| 83 | \( 1 - 8.09iT - 83T^{2} \) |
| 89 | \( 1 + 6.63T + 89T^{2} \) |
| 97 | \( 1 + 0.0268T + 97T^{2} \) |
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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
−10.98639353827496680820798400070, −10.33943654626192179619624703026, −8.764883379196638281644343330446, −7.86012355293524992428175525889, −7.20014047459184929669740022360, −6.37697608386640137577184064997, −5.47983882988932411921994303738, −4.31319833965110758399788669946, −3.18135113610831563772070478327, −2.03662735731724732026012957681,
0.11792702045958484868943438385, 2.85738647096754909888018464216, 3.60322612293362933121710504787, 4.72162862350203429822574320462, 5.32685865611521414766637810313, 6.32120599763338714512450622069, 7.67796753307667480870531382491, 8.207455096976533317241790214302, 9.626405485811967403749656000946, 10.17356285233114527722811365808