| L(s) = 1 | + 2.63i·2-s + 3.04i·3-s − 4.91·4-s − 8.00·6-s + 0.574i·7-s − 7.67i·8-s − 6.26·9-s + 2.57·11-s − 14.9i·12-s − 0.468i·13-s − 1.51·14-s + 10.3·16-s + 4.08i·17-s − 16.4i·18-s − 19-s + ⋯ |
| L(s) = 1 | + 1.85i·2-s + 1.75i·3-s − 2.45·4-s − 3.26·6-s + 0.217i·7-s − 2.71i·8-s − 2.08·9-s + 0.776·11-s − 4.31i·12-s − 0.129i·13-s − 0.403·14-s + 2.58·16-s + 0.991i·17-s − 3.88i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.801557 - 0.495389i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.801557 - 0.495389i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.63iT - 2T^{2} \) |
| 3 | \( 1 - 3.04iT - 3T^{2} \) |
| 7 | \( 1 - 0.574iT - 7T^{2} \) |
| 11 | \( 1 - 2.57T + 11T^{2} \) |
| 13 | \( 1 + 0.468iT - 13T^{2} \) |
| 17 | \( 1 - 4.08iT - 17T^{2} \) |
| 23 | \( 1 - 1.51iT - 23T^{2} \) |
| 29 | \( 1 - 4.08T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 - 8.30iT - 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 + 0.574iT - 43T^{2} \) |
| 47 | \( 1 + 7.09iT - 47T^{2} \) |
| 53 | \( 1 - 4.30iT - 53T^{2} \) |
| 59 | \( 1 - 2.68T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.70iT - 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 6.68T + 79T^{2} \) |
| 83 | \( 1 + 6.66iT - 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 17.4iT - 97T^{2} \) |
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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
−11.56943550790452942862017287569, −10.38505169526629969977286611165, −9.699843947273402174562959068370, −8.799002645917382708472894193659, −8.419350022485815739530381846369, −7.04809613564316833595080411954, −5.98079262484530679692261129863, −5.29890637466141920091595634243, −4.32172223239521193372700528828, −3.60271660837948958690660395966,
0.60555899744086569538930769682, 1.74274653898676888450612137925, 2.68593218287033978848223639891, 3.92336696996106343873576540469, 5.34710853609683234053857021405, 6.66960163328709186276458403143, 7.64832893142091958565772691355, 8.747772527530483109370842127728, 9.380787688922474078329810352621, 10.65618382021046379446636967885
