L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.939 − 0.342i)3-s + (−0.5 − 0.866i)4-s + (0.766 − 0.642i)5-s + (−0.766 + 0.642i)6-s + (0.766 − 0.642i)7-s − 8-s + (0.766 + 0.642i)9-s + (−0.173 − 0.984i)10-s + (−0.173 + 0.984i)11-s + (0.173 + 0.984i)12-s + (−0.766 + 0.642i)13-s + (−0.173 − 0.984i)14-s + (−0.939 + 0.342i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.939 − 0.342i)3-s + (−0.5 − 0.866i)4-s + (0.766 − 0.642i)5-s + (−0.766 + 0.642i)6-s + (0.766 − 0.642i)7-s − 8-s + (0.766 + 0.642i)9-s + (−0.173 − 0.984i)10-s + (−0.173 + 0.984i)11-s + (0.173 + 0.984i)12-s + (−0.766 + 0.642i)13-s + (−0.173 − 0.984i)14-s + (−0.939 + 0.342i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.5004898922 - 0.6900133379i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.5004898922 - 0.6900133379i\) |
\(L(1)\) |
\(\approx\) |
\(0.6437979440 - 0.7358097697i\) |
\(L(1)\) |
\(\approx\) |
\(0.6437979440 - 0.7358097697i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 7 | \( 1 + (0.766 - 0.642i)T \) |
| 11 | \( 1 + (-0.173 + 0.984i)T \) |
| 13 | \( 1 + (-0.766 + 0.642i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.173 + 0.984i)T \) |
| 31 | \( 1 + (-0.939 + 0.342i)T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.766 - 0.642i)T \) |
| 53 | \( 1 + (-0.173 - 0.984i)T \) |
| 59 | \( 1 + (0.939 - 0.342i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (-0.173 + 0.984i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (0.173 - 0.984i)T \) |
| 79 | \( 1 + (-0.173 + 0.984i)T \) |
| 83 | \( 1 + (0.766 - 0.642i)T \) |
| 89 | \( 1 + (-0.939 - 0.342i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.65658391763769384554056948944, −17.948683195896716456283819819997, −17.416678109809861233392896435014, −17.018228384156658796776123023094, −16.28909228068391870750719259515, −15.3741966819694446242854033359, −14.88712831172609172723370657716, −14.51990233948731190155664495550, −13.470339129323533938669680575279, −12.87375183051621352864804826711, −12.226795574238459853685151989282, −11.25005656113045050369887851864, −10.94669815935688600038349032203, −9.88697033301924451747052114921, −9.32312880120849695617579033711, −8.28116174575831147195536154557, −7.72803729460049520458191489162, −6.762976162150581305621566865699, −5.99903827857237033363892843408, −5.69337365967249366761173047682, −5.07576603694998114120113354907, −4.22926526963254563489120105273, −3.2934395316570715782675200135, −2.50877079768018697121567027506, −1.33899841313042258473585882251,
0.22630256533875695082655278275, 1.23774288145527675044743166061, 1.89767090170601810538005275731, 2.42304544143806541636937312351, 3.92056010481532614407372062776, 4.730280785152045137109283995717, 4.97662620915084351321035133166, 5.6206743751616514672963711301, 6.814440326370445286215064004458, 7.09303615797844352883146590494, 8.371890698997027559985208426406, 9.20542326919545332977862731853, 10.035762726237989132076169632092, 10.43837194306969918248649123864, 11.155158831169318435627863368330, 11.99776054247804040906902277137, 12.43128535596463092098086210736, 13.01952937039103598693838122984, 13.701450597475629676120751231178, 14.47559241163892404463731827032, 14.91317422946398969188656756360, 16.26678999416901183723503876924, 16.7146963846522882469467601184, 17.569907957492855961984170559759, 17.87393943015521240363232296283