L(s) = 1 | + (0.913 + 0.406i)2-s + (0.669 − 0.743i)3-s + (0.669 + 0.743i)4-s + (0.809 + 0.587i)5-s + (0.913 − 0.406i)6-s + (0.669 + 0.743i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 + 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (−0.913 + 0.406i)17-s + (0.309 − 0.951i)18-s + (−0.978 − 0.207i)19-s + ⋯ |
L(s) = 1 | + (0.913 + 0.406i)2-s + (0.669 − 0.743i)3-s + (0.669 + 0.743i)4-s + (0.809 + 0.587i)5-s + (0.913 − 0.406i)6-s + (0.669 + 0.743i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 + 0.866i)10-s + 12-s + (0.309 + 0.951i)14-s + (0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (−0.913 + 0.406i)17-s + (0.309 − 0.951i)18-s + (−0.978 − 0.207i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.336230528 + 1.492848163i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.336230528 + 1.492848163i\) |
\(L(1)\) |
\(\approx\) |
\(2.529778794 + 0.5463565294i\) |
\(L(1)\) |
\(\approx\) |
\(2.529778794 + 0.5463565294i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.913 + 0.406i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 + (0.669 + 0.743i)T \) |
| 17 | \( 1 + (-0.913 + 0.406i)T \) |
| 19 | \( 1 + (-0.978 - 0.207i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.978 - 0.207i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (0.978 - 0.207i)T \) |
| 41 | \( 1 + (0.669 - 0.743i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.309 - 0.951i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.669 - 0.743i)T \) |
| 61 | \( 1 + (-0.913 + 0.406i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.913 + 0.406i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.104 + 0.994i)T \) |
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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
−28.0152434969248165810559616962, −27.10272389006780032097464581137, −25.78770417392445861437200086144, −24.90374209205100152975523934104, −24.029296364578027951332402169579, −22.857989324326627039670230304620, −21.55208736218596866489933562761, −21.20762570124627330001443036770, −20.2031624763427654702246264838, −19.57321632482455181392560989696, −17.79272509200581452392606177992, −16.57360451829186960580277269040, −15.563004749909938289468875278054, −14.36363600496174318246196392003, −13.77600874191181416667302318564, −12.84378909636037315849226075748, −11.27165229174094957321976333516, −10.34021878446954401377699324536, −9.37515431141705544440033988274, −7.99060646833749518379749925231, −6.3087366968863266637376733326, −4.85351861517194260685464601568, −4.2915518390933326779492698223, −2.705410406302003390290578831447, −1.46805262446751234410416815392,
2.03401016576059585734597480996, 2.65917566511035328465698685482, 4.358826038389367518560427246033, 5.95810265589855014210604326707, 6.63441052671388592652228953806, 7.978030302360314625149654647156, 8.9379486725567445498510617448, 10.75087121930814326702582532957, 12.013509926265238715360141469062, 13.006073822112396142615576275397, 13.93497960302655225004119889140, 14.72171847364798190292836749337, 15.47336384830386325989800730692, 17.26112059835293787823490290232, 17.97748014318991635455205898757, 19.128006967474582747748440703207, 20.45308459112978797090831748764, 21.35910070341634383466214881533, 22.13019195809136135030995814939, 23.37982588733325106192558967287, 24.4177941447137815750796300021, 24.99329787946153989975377825422, 25.873579401283228458386690377239, 26.66819277597428630781505913623, 28.50264810952901812905230971045