L(s) = 1 | + (0.882 − 0.469i)2-s + (0.559 − 0.829i)4-s + (−0.173 + 0.984i)5-s + (−0.241 + 0.970i)7-s + (0.104 − 0.994i)8-s + (0.309 + 0.951i)10-s + (−0.961 + 0.275i)11-s + (0.0348 + 0.999i)13-s + (0.241 + 0.970i)14-s + (−0.374 − 0.927i)16-s + (0.104 − 0.994i)17-s + (0.309 + 0.951i)19-s + (0.719 + 0.694i)20-s + (−0.719 + 0.694i)22-s + (−0.961 − 0.275i)23-s + ⋯ |
L(s) = 1 | + (0.882 − 0.469i)2-s + (0.559 − 0.829i)4-s + (−0.173 + 0.984i)5-s + (−0.241 + 0.970i)7-s + (0.104 − 0.994i)8-s + (0.309 + 0.951i)10-s + (−0.961 + 0.275i)11-s + (0.0348 + 0.999i)13-s + (0.241 + 0.970i)14-s + (−0.374 − 0.927i)16-s + (0.104 − 0.994i)17-s + (0.309 + 0.951i)19-s + (0.719 + 0.694i)20-s + (−0.719 + 0.694i)22-s + (−0.961 − 0.275i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01780904024 - 0.1735383856i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01780904024 - 0.1735383856i\) |
\(L(1)\) |
\(\approx\) |
\(1.288195520 - 0.07326372999i\) |
\(L(1)\) |
\(\approx\) |
\(1.288195520 - 0.07326372999i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.882 - 0.469i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 7 | \( 1 + (-0.241 + 0.970i)T \) |
| 11 | \( 1 + (-0.961 + 0.275i)T \) |
| 13 | \( 1 + (0.0348 + 0.999i)T \) |
| 17 | \( 1 + (0.104 - 0.994i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
| 23 | \( 1 + (-0.961 - 0.275i)T \) |
| 29 | \( 1 + (0.882 - 0.469i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.615 + 0.788i)T \) |
| 43 | \( 1 + (-0.882 + 0.469i)T \) |
| 47 | \( 1 + (0.615 - 0.788i)T \) |
| 53 | \( 1 + (0.104 - 0.994i)T \) |
| 59 | \( 1 + (-0.0348 - 0.999i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.809 + 0.587i)T \) |
| 73 | \( 1 + (-0.104 - 0.994i)T \) |
| 79 | \( 1 + (0.559 + 0.829i)T \) |
| 83 | \( 1 + (-0.848 - 0.529i)T \) |
| 89 | \( 1 + (-0.913 - 0.406i)T \) |
| 97 | \( 1 + (-0.241 + 0.970i)T \) |
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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
−22.41090494710210844463656296468, −21.53527887735685542325627064811, −20.77968632871489772053245510518, −20.08022025435122183879724898081, −19.56962447405198724327411548559, −17.98662478829321615286996777339, −17.26804374476140508746816041137, −16.5870263882708503956654210103, −15.710541362618398393164584252721, −15.346461455334843233433401379608, −13.944763626826196350309940288303, −13.44659120451883612032749637527, −12.74067472667370211906095131773, −12.07628668678113183594873073136, −10.86781579094870859650108445157, −10.20766117774149622388761677857, −8.77136279175136280797552330991, −7.97501663881136433874483265198, −7.386278983590777794869883553109, −6.1711124531068402080570072095, −5.35635696780216107319070421496, −4.559168376903821857689770474186, −3.6920390377977393215754325321, −2.74871284628426120904373255747, −1.246359468223173604007915248637,
0.025973203432225034603527986281, 1.89933409229880036296263541983, 2.576175767043000156634665142471, 3.39582226207384950739614115123, 4.47441498283120152114349521991, 5.48680366888316104147647447740, 6.274106950971449308620222060988, 7.08942771457817077611113365308, 8.13264652699279991723894904292, 9.57630846514996335596120542047, 10.10613547000271944765945528867, 11.13079760975174094218495543858, 11.87779527687662969728927432702, 12.41207493903791052997526242751, 13.593572230999430725744159664449, 14.22506265145169737244370700671, 14.95957591316548593426973168930, 15.83816562725044083769582521866, 16.25217199032625130793432799909, 18.07460923610454699977128655205, 18.47973884695144173294669803262, 19.161132047678868710582707724917, 20.03550803904715061495680829104, 21.1165119622732562909244036149, 21.50592012980208515880968805678