L(s) = 1 | + (0.990 + 0.139i)2-s + (−0.681 + 0.731i)3-s + (0.960 + 0.276i)4-s + (0.999 + 0.00737i)5-s + (−0.777 + 0.629i)6-s + (0.674 + 0.738i)7-s + (0.912 + 0.408i)8-s + (−0.0709 − 0.997i)9-s + (0.989 + 0.147i)10-s + (−0.976 − 0.217i)11-s + (−0.857 + 0.514i)12-s + (0.980 + 0.194i)13-s + (0.565 + 0.825i)14-s + (−0.686 + 0.726i)15-s + (0.846 + 0.531i)16-s + (0.923 + 0.384i)17-s + ⋯ |
L(s) = 1 | + (0.990 + 0.139i)2-s + (−0.681 + 0.731i)3-s + (0.960 + 0.276i)4-s + (0.999 + 0.00737i)5-s + (−0.777 + 0.629i)6-s + (0.674 + 0.738i)7-s + (0.912 + 0.408i)8-s + (−0.0709 − 0.997i)9-s + (0.989 + 0.147i)10-s + (−0.976 − 0.217i)11-s + (−0.857 + 0.514i)12-s + (0.980 + 0.194i)13-s + (0.565 + 0.825i)14-s + (−0.686 + 0.726i)15-s + (0.846 + 0.531i)16-s + (0.923 + 0.384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.465882640 + 2.934132352i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.465882640 + 2.934132352i\) |
\(L(1)\) |
\(\approx\) |
\(1.864426166 + 0.9739664014i\) |
\(L(1)\) |
\(\approx\) |
\(1.864426166 + 0.9739664014i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3407 | \( 1 \) |
good | 2 | \( 1 + (0.990 + 0.139i)T \) |
| 3 | \( 1 + (-0.681 + 0.731i)T \) |
| 5 | \( 1 + (0.999 + 0.00737i)T \) |
| 7 | \( 1 + (0.674 + 0.738i)T \) |
| 11 | \( 1 + (-0.976 - 0.217i)T \) |
| 13 | \( 1 + (0.980 + 0.194i)T \) |
| 17 | \( 1 + (0.923 + 0.384i)T \) |
| 19 | \( 1 + (0.0212 + 0.999i)T \) |
| 23 | \( 1 + (-0.875 + 0.482i)T \) |
| 29 | \( 1 + (-0.748 + 0.663i)T \) |
| 31 | \( 1 + (0.373 - 0.927i)T \) |
| 37 | \( 1 + (0.826 - 0.562i)T \) |
| 41 | \( 1 + (-0.631 + 0.775i)T \) |
| 43 | \( 1 + (0.390 - 0.920i)T \) |
| 47 | \( 1 + (-0.753 - 0.657i)T \) |
| 53 | \( 1 + (0.0838 + 0.996i)T \) |
| 59 | \( 1 + (0.973 - 0.226i)T \) |
| 61 | \( 1 + (-0.969 - 0.246i)T \) |
| 67 | \( 1 + (-0.839 - 0.542i)T \) |
| 71 | \( 1 + (-0.966 - 0.257i)T \) |
| 73 | \( 1 + (0.437 - 0.899i)T \) |
| 79 | \( 1 + (0.250 + 0.968i)T \) |
| 83 | \( 1 + (-0.392 - 0.919i)T \) |
| 89 | \( 1 + (-0.104 - 0.994i)T \) |
| 97 | \( 1 + (-0.191 + 0.981i)T \) |
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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
−18.39306692092976487639882278358, −17.969416231399704958455198118480, −17.28549045846972920586114061264, −16.45604742975400490549045932468, −16.00642057218677194662329800465, −14.94036041547473453094991886082, −14.13981133652815837878487582249, −13.603681116682183639192042197, −13.177091421349730963134835206054, −12.56754145891142813344577546976, −11.60514763095013089999409531468, −11.09115263209561098807297733675, −10.37344523707045293865768568230, −9.925430147214051267936962055740, −8.41216319737425970903956969318, −7.69006151606273117852228922832, −7.01266587232550062559574080428, −6.24727598269137276174748987631, −5.6237802335975963906702801070, −5.00721453544701238181633921538, −4.37043440949146132519271466861, −3.107862840933456863140908972492, −2.34517972234969848388772202657, −1.53982323288308275668787260715, −0.861194939051877795690324479116,
1.362618787441584124720192336287, 2.042432082072636361548276952161, 3.091193559410597159725887928523, 3.80146214616629202159380443634, 4.73132476369843807951454966256, 5.48595667485210079687387240976, 5.85048489308995476103936201557, 6.25094503954526597707664807068, 7.57741557136399608799032868472, 8.28345320869031524262584594456, 9.23203348966079854191450703502, 10.17961270690002950506556114279, 10.62055305509080141994329669415, 11.402164149321559473893827876, 12.017994660692656042702137131179, 12.75929363632291523254099471442, 13.46549836411169205524738980569, 14.24951498644805862087521832928, 14.840641307842816923517885866698, 15.452385537764756984496524512763, 16.27103465824984381293656724187, 16.65842650946579550579630913651, 17.46358821803050967388225174163, 18.37109652372616548789478044508, 18.584379977081247913093833806284