L(s) = 1 | + (−0.202 − 0.979i)2-s + (−0.917 + 0.396i)4-s + (−0.714 + 0.699i)5-s + (0.557 + 0.830i)7-s + (0.574 + 0.818i)8-s + (0.830 + 0.557i)10-s + (0.242 + 0.970i)11-s + (0.862 − 0.505i)13-s + (0.699 − 0.714i)14-s + (0.685 − 0.728i)16-s + (0.755 − 0.654i)17-s + (−0.396 − 0.917i)19-s + (0.377 − 0.925i)20-s + (0.900 − 0.433i)22-s + (0.0203 − 0.999i)25-s + (−0.670 − 0.742i)26-s + ⋯ |
L(s) = 1 | + (−0.202 − 0.979i)2-s + (−0.917 + 0.396i)4-s + (−0.714 + 0.699i)5-s + (0.557 + 0.830i)7-s + (0.574 + 0.818i)8-s + (0.830 + 0.557i)10-s + (0.242 + 0.970i)11-s + (0.862 − 0.505i)13-s + (0.699 − 0.714i)14-s + (0.685 − 0.728i)16-s + (0.755 − 0.654i)17-s + (−0.396 − 0.917i)19-s + (0.377 − 0.925i)20-s + (0.900 − 0.433i)22-s + (0.0203 − 0.999i)25-s + (−0.670 − 0.742i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.114730639 - 0.5887414167i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.114730639 - 0.5887414167i\) |
\(L(1)\) |
\(\approx\) |
\(0.8727276814 - 0.2589975435i\) |
\(L(1)\) |
\(\approx\) |
\(0.8727276814 - 0.2589975435i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.202 - 0.979i)T \) |
| 5 | \( 1 + (-0.714 + 0.699i)T \) |
| 7 | \( 1 + (0.557 + 0.830i)T \) |
| 11 | \( 1 + (0.242 + 0.970i)T \) |
| 13 | \( 1 + (0.862 - 0.505i)T \) |
| 17 | \( 1 + (0.755 - 0.654i)T \) |
| 19 | \( 1 + (-0.396 - 0.917i)T \) |
| 31 | \( 1 + (-0.872 - 0.488i)T \) |
| 37 | \( 1 + (0.994 + 0.101i)T \) |
| 41 | \( 1 + (0.540 - 0.841i)T \) |
| 43 | \( 1 + (0.872 - 0.488i)T \) |
| 47 | \( 1 + (0.781 - 0.623i)T \) |
| 53 | \( 1 + (-0.996 - 0.0815i)T \) |
| 59 | \( 1 + (0.142 - 0.989i)T \) |
| 61 | \( 1 + (-0.162 - 0.986i)T \) |
| 67 | \( 1 + (-0.970 - 0.242i)T \) |
| 71 | \( 1 + (-0.452 + 0.891i)T \) |
| 73 | \( 1 + (-0.470 + 0.882i)T \) |
| 79 | \( 1 + (0.728 - 0.685i)T \) |
| 83 | \( 1 + (0.992 - 0.122i)T \) |
| 89 | \( 1 + (0.998 - 0.0611i)T \) |
| 97 | \( 1 + (0.852 - 0.523i)T \) |
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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
−19.84426716654298378794688072811, −19.245636754800971365984845648877, −18.59031691989090649815398978718, −17.732601965275285968503504494909, −16.842093844264898497268010617067, −16.43062727535228274024848773442, −16.038004721352842649303146148562, −14.830552704762069940697233676137, −14.43243551538542237744526489707, −13.575557821162995923458597271133, −12.93699293007812034215102672578, −11.99858177997478007813868777920, −11.01067641376871792359409120067, −10.47416313609332722327245086147, −9.24527564419374916338659597709, −8.70683670685716642309111851962, −7.82550042481593820416875600193, −7.62544953001992927059349909389, −6.2981345057066498799871172073, −5.81008449602600014971845066566, −4.71870053420024302425887298066, −4.030584735683480264658459624129, −3.49973966524337798687121096729, −1.425134996976448647881564969118, −0.94146339736478801579846097142,
0.64853709650606050719464348222, 1.91257626016416933920139930370, 2.61574591522265853885514658585, 3.45741920444317692525026049812, 4.29149437969964785739563565857, 5.08421509954793108963254771580, 6.06759555689761974790040043252, 7.32890970309601657389189013016, 7.83780766923029163316480279161, 8.777284996723844250141257583553, 9.39221434110742761761312701725, 10.35690469689436442305292471656, 11.090378250408057556904177335732, 11.568557858829622153647014357598, 12.333569280889335786842764527469, 12.91993622146063286020469106820, 14.01237917216050722981443457540, 14.67038983607006065392253516222, 15.34940312459006904931456116130, 16.104963345772756975613518288062, 17.33192819855821684190194782448, 17.86095601790037534659116825843, 18.611532095357891147341800496245, 18.92539946776672326057570120832, 19.94520043679775596031602299068