L(s) = 1 | + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.499i)6-s + (−1.5 − 2.59i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.232 + 0.133i)11-s − 0.999i·12-s + (−3.5 + 0.866i)13-s − 3·14-s + (−0.5 + 0.866i)16-s + (−3.46 + 2i)17-s + 0.999·18-s + (−4.96 + 2.86i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.353 − 0.204i)6-s + (−0.566 − 0.981i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.0699 + 0.0403i)11-s − 0.288i·12-s + (−0.970 + 0.240i)13-s − 0.801·14-s + (−0.125 + 0.216i)16-s + (−0.840 + 0.485i)17-s + 0.235·18-s + (−1.13 + 0.657i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3.5 - 0.866i)T \) |
good | 7 | \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.232 - 0.133i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (3.46 - 2i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.96 - 2.86i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 1.73i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.732 + 1.26i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4.92iT - 31T^{2} \) |
| 37 | \( 1 + (2.96 - 5.13i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.46 + 2i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.19 + 3i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6.46T + 47T^{2} \) |
| 53 | \( 1 + 0.267iT - 53T^{2} \) |
| 59 | \( 1 + (-9.92 + 5.73i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.267 - 0.464i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.732 + 1.26i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (11.1 - 6.46i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 3.07T + 79T^{2} \) |
| 83 | \( 1 + 9.46T + 83T^{2} \) |
| 89 | \( 1 + (-12.2 - 7.06i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.19 + 7.26i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.766207301364482315873146328641, −8.085245543707413052254221912713, −6.96154377683027520326127900211, −6.44113201758756424590034046968, −5.18249385517977865122126927115, −4.24347609946144078514683546505, −3.80851351160591977331227038068, −2.68597898394885858123676013262, −1.73702970571412030960038402410, 0,
2.21253644858914769900581490115, 2.81334936367720578112579217090, 4.00047488242436642584775203610, 4.89653276676034477975892816512, 5.82287036279973228969756087985, 6.55258345994619908649293073884, 7.25219203941351670009969360330, 8.084319243331291549991132174673, 8.897022534981356196605740374305