L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s − 7-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)12-s + (−0.809 − 0.587i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + 18-s + (−0.309 + 0.951i)21-s + (0.309 − 0.951i)22-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s − 7-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)12-s + (−0.809 − 0.587i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + 18-s + (−0.309 + 0.951i)21-s + (0.309 − 0.951i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3551614605 + 0.1952516652i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3551614605 + 0.1952516652i\) |
\(L(1)\) |
\(\approx\) |
\(0.5541968527 - 0.07546847661i\) |
\(L(1)\) |
\(\approx\) |
\(0.5541968527 - 0.07546847661i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (-0.809 + 0.587i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.809 - 0.587i)T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.309 + 0.951i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.809 + 0.587i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + (0.309 + 0.951i)T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
| 83 | \( 1 + (-0.309 - 0.951i)T \) |
| 89 | \( 1 + (0.809 - 0.587i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−23.26809478159050522084067454590, −22.2269588173357638793455341691, −21.56412942532194126111370929376, −21.0106554086126442648770780446, −19.86279049433994773610712193142, −19.39343025750637566928321053514, −18.63410192710956286676429595807, −17.24919595932670452681669368520, −16.7275798144535193478027636458, −15.83412359524497943987489716129, −15.18791330881746830026024117659, −13.76441055137040731065953004341, −12.94918445430718068885549533773, −11.88026932870548347776937117969, −10.82457096283538304166886254964, −10.20647086075142660796541129247, −9.37116461894482523326726009999, −8.67441577472125123075072892462, −7.64774768256051180100213568776, −6.47161008592534594499446913942, −5.11117856383870688624466087019, −3.81875390567434886349455512062, −3.09582090511598366002783723934, −2.088884355763432058311559441966, −0.19533147685149017583489033813,
0.683414315536152021101579246816, 2.20636769694545850605581378865, 2.980541825288709177257052279807, 4.978490432094648701033529216220, 5.955338895246307188780665140012, 7.105847743541105274449821258617, 7.3607396975519041602678246950, 8.56013359872224404290628976440, 9.43182627671284761812406125596, 10.23649845345825951783742110883, 11.411860662544204097536212395, 12.63526362175109126672507303947, 13.20920138863315928664766299456, 14.40011701889764821951911033735, 15.15525502196521418545415355739, 16.09915695639711558020681225583, 16.99462024371648327589265224432, 17.93672171114328917017631463207, 18.49049790993557592098361966535, 19.36338596099626823259052639242, 20.01977344233998397384520277780, 20.74959882744592826239622323496, 22.563670733130070843168020438821, 22.98164270184238817179380781516, 24.0186248426434684279519400717