L(s) = 1 | + (0.406 + 0.913i)2-s + (0.669 + 0.743i)3-s + (−0.669 + 0.743i)4-s + (−0.587 − 0.809i)5-s + (−0.406 + 0.913i)6-s + (−0.743 − 0.669i)7-s + (−0.951 − 0.309i)8-s + (−0.104 + 0.994i)9-s + (0.5 − 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (0.207 − 0.978i)15-s + (−0.104 − 0.994i)16-s + (−0.913 − 0.406i)17-s + (−0.951 + 0.309i)18-s + (−0.207 − 0.978i)19-s + ⋯ |
L(s) = 1 | + (0.406 + 0.913i)2-s + (0.669 + 0.743i)3-s + (−0.669 + 0.743i)4-s + (−0.587 − 0.809i)5-s + (−0.406 + 0.913i)6-s + (−0.743 − 0.669i)7-s + (−0.951 − 0.309i)8-s + (−0.104 + 0.994i)9-s + (0.5 − 0.866i)10-s − 12-s + (0.309 − 0.951i)14-s + (0.207 − 0.978i)15-s + (−0.104 − 0.994i)16-s + (−0.913 − 0.406i)17-s + (−0.951 + 0.309i)18-s + (−0.207 − 0.978i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3359647830 - 0.2493537555i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3359647830 - 0.2493537555i\) |
\(L(1)\) |
\(\approx\) |
\(0.8313497779 + 0.4185366282i\) |
\(L(1)\) |
\(\approx\) |
\(0.8313497779 + 0.4185366282i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.406 + 0.913i)T \) |
| 3 | \( 1 + (0.669 + 0.743i)T \) |
| 5 | \( 1 + (-0.587 - 0.809i)T \) |
| 7 | \( 1 + (-0.743 - 0.669i)T \) |
| 17 | \( 1 + (-0.913 - 0.406i)T \) |
| 19 | \( 1 + (-0.207 - 0.978i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.978 - 0.207i)T \) |
| 31 | \( 1 + (0.587 - 0.809i)T \) |
| 37 | \( 1 + (-0.207 + 0.978i)T \) |
| 41 | \( 1 + (-0.743 + 0.669i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.951 - 0.309i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.743 - 0.669i)T \) |
| 61 | \( 1 + (0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.406 + 0.913i)T \) |
| 73 | \( 1 + (0.951 - 0.309i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.587 - 0.809i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (-0.994 - 0.104i)T \) |
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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
−28.575044686863720418969570436040, −27.29264918469541194686150545566, −26.353329279277101410991736199659, −25.30999981916691936809606894392, −24.11970910941481536897145466324, −23.13981469912555030620109967136, −22.36118235574422673251786023855, −21.26641183584738523881036197066, −20.03347239465630929750321355522, −19.22842854826620400328390910014, −18.741533218333800175219357902671, −17.71726412571428260230297389475, −15.590203054592571882323665039560, −14.82675190843296600343697300351, −13.79078186766941480454603722073, −12.73464198589996151428651817361, −11.97252592821130634475285207593, −10.79823001029214579436257031715, −9.490882167740018254226569648690, −8.44924407598243197724198930570, −6.96121122916614972454420913693, −5.85257269055518455671011941036, −3.84724404384703655844681814433, −3.00722730235881061889002160557, −1.868629996200915505125808405887,
0.1236485172522824010855736826, 3.03105678807952213694121927128, 4.22022197818199402520608938553, 4.89610092128799216788900457219, 6.63104406200697042787986297707, 7.8286832442745589106809699968, 8.82996702778886707921572947128, 9.68351434036711909260500153645, 11.334633291044100758277299840641, 12.997203654564943882710503318101, 13.468813745218353418744196167071, 14.86405920178391242151434012965, 15.68841907274340003564605782756, 16.443713902904415869464682359793, 17.23307967283064810201955939592, 18.988422940168580029311974876419, 20.081135428655274296299755020189, 20.84439919782662389607350365248, 22.135840388354856743290390418456, 22.88963074783699344740870694152, 24.07270323234716021214764235880, 24.84626312744346113813235058423, 26.015267426270713310524019920971, 26.59891314115334602641405504592, 27.51487492948757709154448209730