L(s) = 1 | + (0.466 + 0.884i)2-s + (−0.309 − 0.951i)3-s + (−0.564 + 0.825i)4-s + (0.696 − 0.717i)6-s + (0.254 + 0.967i)7-s + (−0.993 − 0.113i)8-s + (−0.809 + 0.587i)9-s + (0.959 + 0.281i)12-s + (−0.941 + 0.336i)13-s + (−0.736 + 0.676i)14-s + (−0.362 − 0.931i)16-s + (−0.974 + 0.226i)17-s + (−0.897 − 0.441i)18-s + (0.516 − 0.856i)19-s + (0.841 − 0.540i)21-s + ⋯ |
L(s) = 1 | + (0.466 + 0.884i)2-s + (−0.309 − 0.951i)3-s + (−0.564 + 0.825i)4-s + (0.696 − 0.717i)6-s + (0.254 + 0.967i)7-s + (−0.993 − 0.113i)8-s + (−0.809 + 0.587i)9-s + (0.959 + 0.281i)12-s + (−0.941 + 0.336i)13-s + (−0.736 + 0.676i)14-s + (−0.362 − 0.931i)16-s + (−0.974 + 0.226i)17-s + (−0.897 − 0.441i)18-s + (0.516 − 0.856i)19-s + (0.841 − 0.540i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.05178348162 + 0.1513112676i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.05178348162 + 0.1513112676i\) |
\(L(1)\) |
\(\approx\) |
\(0.7329993168 + 0.2872031614i\) |
\(L(1)\) |
\(\approx\) |
\(0.7329993168 + 0.2872031614i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.466 + 0.884i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.254 + 0.967i)T \) |
| 13 | \( 1 + (-0.941 + 0.336i)T \) |
| 17 | \( 1 + (-0.974 + 0.226i)T \) |
| 19 | \( 1 + (0.516 - 0.856i)T \) |
| 23 | \( 1 + (-0.841 - 0.540i)T \) |
| 29 | \( 1 + (-0.921 + 0.389i)T \) |
| 31 | \( 1 + (0.610 - 0.791i)T \) |
| 37 | \( 1 + (0.0285 - 0.999i)T \) |
| 41 | \( 1 + (-0.985 + 0.170i)T \) |
| 43 | \( 1 + (-0.415 + 0.909i)T \) |
| 47 | \( 1 + (-0.897 + 0.441i)T \) |
| 53 | \( 1 + (0.362 - 0.931i)T \) |
| 59 | \( 1 + (-0.985 - 0.170i)T \) |
| 61 | \( 1 + (-0.466 + 0.884i)T \) |
| 67 | \( 1 + (0.142 - 0.989i)T \) |
| 71 | \( 1 + (0.0855 + 0.996i)T \) |
| 73 | \( 1 + (-0.774 + 0.633i)T \) |
| 79 | \( 1 + (-0.870 - 0.491i)T \) |
| 83 | \( 1 + (0.998 + 0.0570i)T \) |
| 89 | \( 1 + (-0.654 + 0.755i)T \) |
| 97 | \( 1 + (-0.198 - 0.980i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.37581622261254305783890583030, −21.84397072431084810411049456483, −20.877735367879001470806028534867, −20.185436309241517059338911573052, −19.78703348224490133175111497309, −18.46312823204168326899715168941, −17.5278324752193191873700721613, −16.87020513087725091430990998662, −15.69660457800354906969776686841, −14.933225466608314982836027523719, −14.064787449753686066089001998370, −13.38724249342701680821653714561, −12.0878428062543704749285650397, −11.54781744846423336921150043430, −10.48134354764508860115870649989, −10.0856240364059725268266097935, −9.20559042442083733387044985637, −7.99129749905778241354032358626, −6.617000708143964075482790488, −5.44731701873356468166064474568, −4.70276964162494473428632394095, −3.89756260025189081929297581296, −3.03618277192045255569978675444, −1.652972903500127874558739093040, −0.06735909342285364513548780969,
2.00682927891413294545904218631, 2.872163855488119596088460193733, 4.48684311843270950401741262917, 5.29799995754650152006561903698, 6.185649830588390336551520600554, 6.940810556539849782292613975617, 7.8409175024954597473718238791, 8.64716367107000078003037505450, 9.5384261127040513739797706118, 11.273409638985359095384803048477, 11.91954878949686043281457670369, 12.73111329709592251582190075114, 13.43416463721342565154687293198, 14.409166901660675446129436419234, 15.08662396405498926448470568932, 16.06869809722261885386656150098, 16.9523255816016205882672110682, 17.80699902169273966964926992066, 18.26302729987524579980598423418, 19.24303704411974563549468607244, 20.206765303392613959077475625908, 21.54700894730641152885429805849, 22.15350476954788572177498786778, 22.74240891429211939823656352057, 23.88448454367605983416747841261