L(s) = 1 | + (0.995 − 0.0980i)3-s + (0.471 − 0.881i)5-s + (0.980 − 0.195i)9-s + (0.773 − 0.634i)11-s + (0.881 − 0.471i)13-s + (0.382 − 0.923i)15-s + (−0.382 − 0.923i)17-s + (0.956 + 0.290i)19-s + (−0.831 − 0.555i)23-s + (−0.555 − 0.831i)25-s + (0.956 − 0.290i)27-s + (−0.634 + 0.773i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (0.290 + 0.956i)37-s + ⋯ |
L(s) = 1 | + (0.995 − 0.0980i)3-s + (0.471 − 0.881i)5-s + (0.980 − 0.195i)9-s + (0.773 − 0.634i)11-s + (0.881 − 0.471i)13-s + (0.382 − 0.923i)15-s + (−0.382 − 0.923i)17-s + (0.956 + 0.290i)19-s + (−0.831 − 0.555i)23-s + (−0.555 − 0.831i)25-s + (0.956 − 0.290i)27-s + (−0.634 + 0.773i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (0.290 + 0.956i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.363666358 - 3.282106362i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.363666358 - 3.282106362i\) |
\(L(1)\) |
\(\approx\) |
\(1.808304688 - 0.6529779202i\) |
\(L(1)\) |
\(\approx\) |
\(1.808304688 - 0.6529779202i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.995 - 0.0980i)T \) |
| 5 | \( 1 + (0.471 - 0.881i)T \) |
| 11 | \( 1 + (0.773 - 0.634i)T \) |
| 13 | \( 1 + (0.881 - 0.471i)T \) |
| 17 | \( 1 + (-0.382 - 0.923i)T \) |
| 19 | \( 1 + (0.956 + 0.290i)T \) |
| 23 | \( 1 + (-0.831 - 0.555i)T \) |
| 29 | \( 1 + (-0.634 + 0.773i)T \) |
| 31 | \( 1 + (0.707 + 0.707i)T \) |
| 37 | \( 1 + (0.290 + 0.956i)T \) |
| 41 | \( 1 + (0.555 - 0.831i)T \) |
| 43 | \( 1 + (0.995 + 0.0980i)T \) |
| 47 | \( 1 + (0.923 - 0.382i)T \) |
| 53 | \( 1 + (-0.634 - 0.773i)T \) |
| 59 | \( 1 + (0.881 + 0.471i)T \) |
| 61 | \( 1 + (-0.0980 - 0.995i)T \) |
| 67 | \( 1 + (-0.0980 - 0.995i)T \) |
| 71 | \( 1 + (-0.980 - 0.195i)T \) |
| 73 | \( 1 + (0.195 + 0.980i)T \) |
| 79 | \( 1 + (0.923 + 0.382i)T \) |
| 83 | \( 1 + (-0.290 + 0.956i)T \) |
| 89 | \( 1 + (0.831 - 0.555i)T \) |
| 97 | \( 1 + (-0.707 - 0.707i)T \) |
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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
−20.28192889933419861717553372451, −19.27601558957846051543962175432, −19.02057749620477991499468929994, −17.90413434097206555269858796330, −17.59212560601989733157017752400, −16.36931981672981217723351954177, −15.56315527247364425534901764313, −14.94522212865753303073823078126, −14.26419474196235427312082137845, −13.66261189871518837313357649735, −13.05720930385075647368853381614, −11.93672709346419300359964566681, −11.12433782864540729719999844864, −10.29313496364018831646114888508, −9.468293430273576556202678612664, −9.10420181757551185229496347815, −7.91399604715078419066857428677, −7.34527626151858941602867570696, −6.414836380183344857517918728882, −5.78264552349041743276743165219, −4.20101487653264607468626035974, −3.89983430464356659807135178439, −2.79520870214279049514452685191, −2.02428217448696988084956719875, −1.25173897862257246298269889377,
0.74727786456127651044334639195, 1.35177069229901466928139878039, 2.40209972820689123155462186966, 3.36448944730839711241517585994, 4.1001770928059131475773817198, 5.07812071055284572007887891972, 5.98238787902672528536823934891, 6.84032513376730939637369560752, 7.879729109455275151090191507853, 8.54997953503415154797524735985, 9.13832892373061494273992345404, 9.76153890805755400029442943160, 10.710064144603977353886720208478, 11.78519596031879428419248188835, 12.50245730065072742078735136449, 13.32200509261216045442417227224, 13.95467202515031160749067645416, 14.30678334360674741216672790250, 15.60510608701261549701521703773, 16.03037315442482124938615934665, 16.75573748001897008892732260630, 17.84434441935358972159251362307, 18.327107738006089798737692140871, 19.22202931599827555845109276302, 20.04722104986708466302631573310