L(s) = 1 | − 0.449i·2-s − 1.95i·3-s + 1.79·4-s + (1.87 − 1.21i)5-s − 0.879·6-s − 2.06i·7-s − 1.70i·8-s − 0.829·9-s + (−0.545 − 0.843i)10-s − 2.30·11-s − 3.51i·12-s + 4.39i·13-s − 0.926·14-s + (−2.37 − 3.67i)15-s + 2.82·16-s − 5.47i·17-s + ⋯ |
L(s) = 1 | − 0.317i·2-s − 1.12i·3-s + 0.899·4-s + (0.839 − 0.543i)5-s − 0.358·6-s − 0.778i·7-s − 0.603i·8-s − 0.276·9-s + (−0.172 − 0.266i)10-s − 0.695·11-s − 1.01i·12-s + 1.21i·13-s − 0.247·14-s + (−0.613 − 0.948i)15-s + 0.707·16-s − 1.32i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.464306435\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.464306435\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.87 + 1.21i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 0.449iT - 2T^{2} \) |
| 3 | \( 1 + 1.95iT - 3T^{2} \) |
| 7 | \( 1 + 2.06iT - 7T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 - 4.39iT - 13T^{2} \) |
| 17 | \( 1 + 5.47iT - 17T^{2} \) |
| 23 | \( 1 - 5.81iT - 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 - 0.757T + 31T^{2} \) |
| 37 | \( 1 + 6.22iT - 37T^{2} \) |
| 41 | \( 1 + 6.53T + 41T^{2} \) |
| 43 | \( 1 + 3.16iT - 43T^{2} \) |
| 47 | \( 1 - 6.36iT - 47T^{2} \) |
| 53 | \( 1 - 3.85iT - 53T^{2} \) |
| 59 | \( 1 + 2.55T + 59T^{2} \) |
| 61 | \( 1 - 9.94T + 61T^{2} \) |
| 67 | \( 1 - 1.70iT - 67T^{2} \) |
| 71 | \( 1 - 9.85T + 71T^{2} \) |
| 73 | \( 1 + 10.2iT - 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 - 7.06iT - 83T^{2} \) |
| 89 | \( 1 - 2.33T + 89T^{2} \) |
| 97 | \( 1 - 6.81iT - 97T^{2} \) |
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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
−9.104791271596382362975713636458, −7.81624499890877200730023832285, −7.28899940430278814905988362929, −6.75878406697075473042058761633, −5.90619016946482017107663163542, −4.98558851410264859297739604016, −3.75919373703157531814658310959, −2.43738139555461041002808205224, −1.81011624613225824394803583230, −0.871970624407079741108251061909,
1.88870343375113015251439298290, 2.78132394313188499533630615742, 3.56474887933259528690918466175, 4.99976243869427058471627023951, 5.59235588276078512923874262929, 6.21996496238737652587139927460, 7.10275740684845394853455412471, 8.165592439466195698898204766151, 8.717338758930858076191090770012, 10.00334270939216669615285292771