L(s) = 1 | + (0.928 − 0.371i)2-s + (−0.5 + 0.866i)3-s + (0.723 − 0.690i)4-s + (0.580 − 0.814i)5-s + (−0.142 + 0.989i)6-s + (0.415 − 0.909i)8-s + (−0.5 − 0.866i)9-s + (0.235 − 0.971i)10-s + (0.235 + 0.971i)12-s + (−0.959 − 0.281i)13-s + (0.415 + 0.909i)15-s + (0.0475 − 0.998i)16-s + (−0.327 − 0.945i)17-s + (−0.786 − 0.618i)18-s + (−0.327 + 0.945i)19-s + (−0.142 − 0.989i)20-s + ⋯ |
L(s) = 1 | + (0.928 − 0.371i)2-s + (−0.5 + 0.866i)3-s + (0.723 − 0.690i)4-s + (0.580 − 0.814i)5-s + (−0.142 + 0.989i)6-s + (0.415 − 0.909i)8-s + (−0.5 − 0.866i)9-s + (0.235 − 0.971i)10-s + (0.235 + 0.971i)12-s + (−0.959 − 0.281i)13-s + (0.415 + 0.909i)15-s + (0.0475 − 0.998i)16-s + (−0.327 − 0.945i)17-s + (−0.786 − 0.618i)18-s + (−0.327 + 0.945i)19-s + (−0.142 − 0.989i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.289763821 - 1.503878369i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.289763821 - 1.503878369i\) |
\(L(1)\) |
\(\approx\) |
\(1.436262463 - 0.5258364063i\) |
\(L(1)\) |
\(\approx\) |
\(1.436262463 - 0.5258364063i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.928 - 0.371i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.580 - 0.814i)T \) |
| 13 | \( 1 + (-0.959 - 0.281i)T \) |
| 17 | \( 1 + (-0.327 - 0.945i)T \) |
| 19 | \( 1 + (-0.327 + 0.945i)T \) |
| 23 | \( 1 + (0.0475 - 0.998i)T \) |
| 29 | \( 1 + (-0.654 - 0.755i)T \) |
| 31 | \( 1 + (0.235 - 0.971i)T \) |
| 37 | \( 1 + (0.723 + 0.690i)T \) |
| 41 | \( 1 + (-0.142 + 0.989i)T \) |
| 43 | \( 1 + (0.415 - 0.909i)T \) |
| 47 | \( 1 + (-0.786 + 0.618i)T \) |
| 53 | \( 1 + (0.0475 + 0.998i)T \) |
| 59 | \( 1 + (0.928 + 0.371i)T \) |
| 61 | \( 1 + (-0.786 + 0.618i)T \) |
| 67 | \( 1 + (-0.786 - 0.618i)T \) |
| 71 | \( 1 + (-0.654 - 0.755i)T \) |
| 73 | \( 1 + (-0.888 + 0.458i)T \) |
| 79 | \( 1 + (0.580 - 0.814i)T \) |
| 83 | \( 1 + (0.841 - 0.540i)T \) |
| 89 | \( 1 + (0.981 + 0.189i)T \) |
| 97 | \( 1 + (0.415 - 0.909i)T \) |
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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
−22.20419774446822350798112029017, −21.92392031789710966020485267686, −21.129513502605077911000845999684, −19.66371642094149053576723431599, −19.38737195074645571648252534921, −18.03625496954674413681220151615, −17.50038422110013394308815185106, −16.87594406847922598233886309411, −15.81885549069481429162027907973, −14.76139992456953708806306636183, −14.310681826370446138226493091185, −13.26822016214811134059303090261, −12.890136506274256680247526669509, −11.832742950621721972872378720326, −11.138213883433582398178591099613, −10.37746895965317917412909846320, −8.981461994966293057569473850314, −7.73901673363517860295872191209, −7.04188923024180641934725187011, −6.46428686133968975232798513356, −5.59238247789526809536456817904, −4.82442029560709596205340569321, −3.47257708610254717007430689657, −2.42486429254896874329909242585, −1.748669621305373657677931315933,
0.64035815818634513730926521344, 2.09349734107331631790030032881, 3.05989650608684281721663272510, 4.44515976838266744915916170877, 4.65833744925140016068004666092, 5.75175359366049095495509085542, 6.25844196876101783840972802390, 7.635020102674109126400013244629, 9.01281485364133588324329939285, 9.84530736533471261375032456109, 10.33490873808311920536673389818, 11.47347100306594618917284653879, 12.10235752372270316834621527672, 12.88145674828041863884164257733, 13.756010534835505098016712039817, 14.70794296369063097593668783121, 15.274935702552100803315638661784, 16.44722253806017806399553566400, 16.668138128950815385953431547586, 17.74074407635838622968714859009, 18.8165172255722747929463331137, 19.99245516466874484843333134621, 20.63238974829222852485958995859, 21.01147206743070095722241237061, 22.027347183614063634402106669340