L(s) = 1 | + (0.957 − 0.286i)2-s + (0.835 − 0.549i)4-s + (0.998 + 0.0581i)7-s + (0.642 − 0.766i)8-s + (−0.597 − 0.802i)11-s + (0.727 − 0.686i)13-s + (0.973 − 0.230i)14-s + (0.396 − 0.918i)16-s + (−0.984 + 0.173i)17-s + (−0.173 + 0.984i)19-s + (−0.802 − 0.597i)22-s + (−0.998 + 0.0581i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.5i)28-s + (0.973 + 0.230i)29-s + ⋯ |
L(s) = 1 | + (0.957 − 0.286i)2-s + (0.835 − 0.549i)4-s + (0.998 + 0.0581i)7-s + (0.642 − 0.766i)8-s + (−0.597 − 0.802i)11-s + (0.727 − 0.686i)13-s + (0.973 − 0.230i)14-s + (0.396 − 0.918i)16-s + (−0.984 + 0.173i)17-s + (−0.173 + 0.984i)19-s + (−0.802 − 0.597i)22-s + (−0.998 + 0.0581i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.5i)28-s + (0.973 + 0.230i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.337725715 - 1.105196708i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.337725715 - 1.105196708i\) |
\(L(1)\) |
\(\approx\) |
\(1.892420624 - 0.5385268888i\) |
\(L(1)\) |
\(\approx\) |
\(1.892420624 - 0.5385268888i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.957 - 0.286i)T \) |
| 7 | \( 1 + (0.998 + 0.0581i)T \) |
| 11 | \( 1 + (-0.597 - 0.802i)T \) |
| 13 | \( 1 + (0.727 - 0.686i)T \) |
| 17 | \( 1 + (-0.984 + 0.173i)T \) |
| 19 | \( 1 + (-0.173 + 0.984i)T \) |
| 23 | \( 1 + (-0.998 + 0.0581i)T \) |
| 29 | \( 1 + (0.973 + 0.230i)T \) |
| 31 | \( 1 + (0.893 + 0.448i)T \) |
| 37 | \( 1 + (-0.342 + 0.939i)T \) |
| 41 | \( 1 + (0.286 - 0.957i)T \) |
| 43 | \( 1 + (-0.116 - 0.993i)T \) |
| 47 | \( 1 + (0.448 + 0.893i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.597 - 0.802i)T \) |
| 61 | \( 1 + (-0.835 - 0.549i)T \) |
| 67 | \( 1 + (-0.230 - 0.973i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.642 + 0.766i)T \) |
| 79 | \( 1 + (0.286 + 0.957i)T \) |
| 83 | \( 1 + (-0.957 + 0.286i)T \) |
| 89 | \( 1 + (0.766 + 0.642i)T \) |
| 97 | \( 1 + (-0.918 - 0.396i)T \) |
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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
−24.263916886208137875411208340101, −23.62608651309520629180695352605, −22.88682098488787405186062539310, −21.80911688895896454977831649183, −21.138531668086787521264083518127, −20.412996473601692800714770110584, −19.53896305829708427767757884636, −17.98313749512023425877220351194, −17.58538426973286169774858648113, −16.269581510552192052166671143582, −15.53949749356875692094357323469, −14.74511131316014156628885177505, −13.78550596917540619407485146470, −13.1577755026303753333453402677, −11.9458439343488342225898033586, −11.30653907812858766589569129248, −10.35398034695002890025993776260, −8.80905037414399732342537893983, −7.87991628711973497244601560320, −6.93247677765795504851038552911, −5.96659525516613939518402632820, −4.64968493524427370247639037066, −4.33468972190392482837759958614, −2.67995422807513089161626106553, −1.76852464931835837533329477088,
1.28114207420602984102517944156, 2.45272899141686074596905763825, 3.60278619065756833735315966773, 4.63601469426550518565466656664, 5.60937968029710346576651037813, 6.410093886218395697947635724986, 7.83783387258799760920636597164, 8.56231269264094050370115211860, 10.303214942098773422176871700076, 10.82733020327060407659151534623, 11.77341465480197154955109823987, 12.65963356919153886481067166230, 13.7622446242161250952810413803, 14.17853895887343276509742939745, 15.48739070102809817933275180074, 15.85035552923617776527338336095, 17.21171548602537840791453806045, 18.23198642927646355234299389919, 19.09517883355595391273403358232, 20.25813639608489805146702895548, 20.819291867530889630479989357693, 21.61086495001127306844014702754, 22.41439133749498055989606076969, 23.44846910989566808858802590064, 24.03221123715182110256144820801