L(s) = 1 | + (−0.960 + 0.279i)2-s + (−0.134 + 0.990i)3-s + (0.843 − 0.536i)4-s + (0.918 − 0.396i)5-s + (−0.147 − 0.989i)6-s + (−0.967 + 0.252i)7-s + (−0.660 + 0.750i)8-s + (−0.963 − 0.266i)9-s + (−0.770 + 0.636i)10-s + (0.418 + 0.908i)12-s + (−0.753 − 0.657i)13-s + (0.858 − 0.512i)14-s + (0.269 + 0.963i)15-s + (0.424 − 0.905i)16-s + (0.796 − 0.604i)17-s + (0.999 − 0.0138i)18-s + ⋯ |
L(s) = 1 | + (−0.960 + 0.279i)2-s + (−0.134 + 0.990i)3-s + (0.843 − 0.536i)4-s + (0.918 − 0.396i)5-s + (−0.147 − 0.989i)6-s + (−0.967 + 0.252i)7-s + (−0.660 + 0.750i)8-s + (−0.963 − 0.266i)9-s + (−0.770 + 0.636i)10-s + (0.418 + 0.908i)12-s + (−0.753 − 0.657i)13-s + (0.858 − 0.512i)14-s + (0.269 + 0.963i)15-s + (0.424 − 0.905i)16-s + (0.796 − 0.604i)17-s + (0.999 − 0.0138i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01968797580 - 0.05578334185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01968797580 - 0.05578334185i\) |
\(L(1)\) |
\(\approx\) |
\(0.5621339140 + 0.1470804462i\) |
\(L(1)\) |
\(\approx\) |
\(0.5621339140 + 0.1470804462i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 547 | \( 1 \) |
good | 2 | \( 1 + (-0.960 + 0.279i)T \) |
| 3 | \( 1 + (-0.134 + 0.990i)T \) |
| 5 | \( 1 + (0.918 - 0.396i)T \) |
| 7 | \( 1 + (-0.967 + 0.252i)T \) |
| 13 | \( 1 + (-0.753 - 0.657i)T \) |
| 17 | \( 1 + (0.796 - 0.604i)T \) |
| 19 | \( 1 + (-0.521 + 0.853i)T \) |
| 23 | \( 1 + (-0.813 - 0.582i)T \) |
| 29 | \( 1 + (0.141 - 0.989i)T \) |
| 31 | \( 1 + (-0.954 - 0.299i)T \) |
| 37 | \( 1 + (0.933 + 0.357i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + (0.978 - 0.205i)T \) |
| 47 | \( 1 + (0.399 - 0.916i)T \) |
| 53 | \( 1 + (0.348 - 0.937i)T \) |
| 59 | \( 1 + (-0.715 + 0.698i)T \) |
| 61 | \( 1 + (-0.997 - 0.0758i)T \) |
| 67 | \( 1 + (-0.418 - 0.908i)T \) |
| 71 | \( 1 + (0.503 + 0.863i)T \) |
| 73 | \( 1 + (0.783 - 0.620i)T \) |
| 79 | \( 1 + (-0.670 + 0.741i)T \) |
| 83 | \( 1 + (-0.262 + 0.964i)T \) |
| 89 | \( 1 + (-0.851 - 0.524i)T \) |
| 97 | \( 1 + (-0.113 + 0.993i)T \) |
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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
−18.00011431989777627315394578268, −17.37597757122974815650483465113, −16.91288794552206365987443792150, −16.38931892501980276956468588314, −15.51090248125363525912245558207, −14.49183827809094717323150899440, −14.0526048919067256741449133337, −13.140488127669375766388676188729, −12.59237444600271312631808649923, −12.202375466381527759694787665188, −11.17365805080172868896344561471, −10.664040966723383204691858402466, −9.99753101511860964039578584918, −9.14184071609473873534246185927, −8.98446320029832235041881276958, −7.60712345919122669193703201809, −7.39983192006538719334322486634, −6.58143303805579215540097350019, −6.1286690280825090708767351206, −5.4560743083998984268469224673, −4.05564985433830093726089962641, −3.06239534467353657509224823079, −2.55738966023227853822621753896, −1.817886189269383718596961814808, −1.14115855132239592097491728100,
0.02403838051550305176378283115, 0.937619737762584019450239119, 2.27010956506974751814858495272, 2.65152913339976599628717034535, 3.60542785633403950243521132733, 4.63481092712494784909422583061, 5.523613135283708883267426354817, 5.90806016267076968915354984353, 6.44770453169341384495344699244, 7.55022341802443805649735059169, 8.24496418833016588185668538723, 8.99955894507054240297108526979, 9.653059634402296572371588767505, 9.959270884236425889169401060800, 10.37822343281004101879800844894, 11.28884218366331793539580268619, 12.20969061288514891437950643054, 12.5899120470436442387741900028, 13.7203759834190110327388315736, 14.42282587836574010138475305093, 15.0192821185962005652040959831, 15.65469541354043458511441439449, 16.4977272423197265969187000629, 16.67001352078223319146872401799, 17.18023014435222276349537334275