L(s) = 1 | + (−0.984 − 0.173i)3-s + (−0.642 + 0.766i)7-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (0.939 + 0.342i)17-s + (0.984 + 0.173i)19-s + (0.766 − 0.642i)21-s + (−0.5 + 0.866i)23-s + (−0.866 − 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (0.342 + 0.939i)33-s + (−0.984 + 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯ |
L(s) = 1 | + (−0.984 − 0.173i)3-s + (−0.642 + 0.766i)7-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.939 − 0.342i)13-s + (0.939 + 0.342i)17-s + (0.984 + 0.173i)19-s + (0.766 − 0.642i)21-s + (−0.5 + 0.866i)23-s + (−0.866 − 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (0.342 + 0.939i)33-s + (−0.984 + 0.173i)39-s + (0.939 − 0.342i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9823990245 + 0.09366644729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9823990245 + 0.09366644729i\) |
\(L(1)\) |
\(\approx\) |
\(0.7863702154 + 0.01101942444i\) |
\(L(1)\) |
\(\approx\) |
\(0.7863702154 + 0.01101942444i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.984 - 0.173i)T \) |
| 7 | \( 1 + (-0.642 + 0.766i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.939 - 0.342i)T \) |
| 17 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (0.984 + 0.173i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 - iT \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.866 - 0.5i)T \) |
| 53 | \( 1 + (0.642 + 0.766i)T \) |
| 59 | \( 1 + (0.642 + 0.766i)T \) |
| 61 | \( 1 + (-0.342 - 0.939i)T \) |
| 67 | \( 1 + (-0.642 + 0.766i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (0.642 - 0.766i)T \) |
| 83 | \( 1 + (0.342 - 0.939i)T \) |
| 89 | \( 1 + (0.642 + 0.766i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−20.75757892589791625432375976926, −20.00644172530126215796261116672, −18.99260044491781960416683236299, −18.181452022770588796921237506195, −17.77529183311194228462357614078, −16.62433910812502032607399047219, −16.328138328807148128405100272463, −15.6450250870199063472558053075, −14.62815061062499358126060167729, −13.66498745607252308491466312174, −12.95271105913471645608254140840, −12.23131560571496602342931065308, −11.43939343545781797422157315391, −10.61559079761791078818118864979, −9.95132976719241484260997823522, −9.39919579370499451363512390576, −8.01414046561076086380715319391, −7.18976810363263134143447021100, −6.54812405583879781417773045466, −5.66627035619888244578774153860, −4.82396192656818253652076192568, −3.98566601197924036286112722939, −3.14278011437279560202167203977, −1.67588437410027272825985722266, −0.671100126327561874909801231843,
0.75090914451359834788633781377, 1.818301323213262842912481449068, 3.153716157787023191196544465522, 3.798211750089820737471151986398, 5.29008715224884113107957471572, 5.71153685149489727988987759234, 6.258480968884894071384630519606, 7.43440333281949455442865628411, 8.12874816417916603036851323592, 9.214477034920446931492545486024, 10.00247385860519839468282099361, 10.79510883707754823995734411064, 11.60688617553526659510569753649, 12.14858230981901503400367942808, 13.172274758882113652929650809237, 13.4695017809250055441775665647, 14.77864776669996208699185910097, 15.716849522338606647646646073059, 16.17844769011574504564178906019, 16.7728086192227966894177168097, 17.835916433808555182346623045892, 18.52598934377912794023699392187, 18.820179353395240770679595276914, 19.826527820834917278414338427546, 20.91458104320161517147102764814