L(s) = 1 | + (0.913 − 0.406i)2-s + (0.669 − 0.743i)3-s + (0.669 − 0.743i)4-s + (0.309 − 0.951i)6-s + (0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)12-s + (−0.809 + 0.587i)13-s + (−0.104 − 0.994i)16-s + (−0.978 + 0.207i)17-s + (−0.5 − 0.866i)18-s + (0.669 + 0.743i)19-s + (0.309 + 0.951i)22-s + (0.913 − 0.406i)23-s + (−0.5 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.913 − 0.406i)2-s + (0.669 − 0.743i)3-s + (0.669 − 0.743i)4-s + (0.309 − 0.951i)6-s + (0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)12-s + (−0.809 + 0.587i)13-s + (−0.104 − 0.994i)16-s + (−0.978 + 0.207i)17-s + (−0.5 − 0.866i)18-s + (0.669 + 0.743i)19-s + (0.309 + 0.951i)22-s + (0.913 − 0.406i)23-s + (−0.5 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.724214182 - 1.427183203i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.724214182 - 1.427183203i\) |
\(L(1)\) |
\(\approx\) |
\(1.730972144 - 0.9144977825i\) |
\(L(1)\) |
\(\approx\) |
\(1.730972144 - 0.9144977825i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.913 - 0.406i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 11 | \( 1 + (-0.104 + 0.994i)T \) |
| 13 | \( 1 + (-0.809 + 0.587i)T \) |
| 17 | \( 1 + (-0.978 + 0.207i)T \) |
| 19 | \( 1 + (0.669 + 0.743i)T \) |
| 23 | \( 1 + (0.913 - 0.406i)T \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.978 + 0.207i)T \) |
| 37 | \( 1 + (-0.104 - 0.994i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.978 - 0.207i)T \) |
| 53 | \( 1 + (0.669 - 0.743i)T \) |
| 59 | \( 1 + (0.913 + 0.406i)T \) |
| 61 | \( 1 + (0.913 - 0.406i)T \) |
| 67 | \( 1 + (-0.978 + 0.207i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.104 + 0.994i)T \) |
| 79 | \( 1 + (-0.978 - 0.207i)T \) |
| 83 | \( 1 + (0.309 - 0.951i)T \) |
| 89 | \( 1 + (0.913 - 0.406i)T \) |
| 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
−27.21285497916599611819083236271, −26.64501580036760548547517596346, −25.63424662433143021335580795885, −24.69951795628989389339771461186, −24.01050584514020564648890585833, −22.510972813895331168282222707916, −22.00825139849316017887333151786, −21.05869949435877007888033614775, −20.16010444876051403197648183734, −19.27833585288688283739033421112, −17.55065593662058203495937600701, −16.50871960788738228737164754380, −15.575599762828475153715547618451, −14.92227189494168795115812472970, −13.76056958287073810439700340710, −13.17296737229100663717960582069, −11.61945360306147631709100825298, −10.70050448082141552053611528668, −9.24538184494304449299172439895, −8.165302157987502729768000533881, −7.08094867388737655364984528411, −5.545003213967157399639985888044, −4.6578694566302039090206665962, −3.377340909931695231051786178130, −2.502389997472929443423803666728,
1.63130441763341347159084756603, 2.58885419825667096792876057463, 3.91836066463791870008498854572, 5.16488893783887237915494031583, 6.70721048315556307478660966468, 7.349332607852746739203598574201, 8.99729289064550953116106387267, 10.100905278224578617267712629187, 11.51816515711024895821575988864, 12.507222325333697408557327319760, 13.131522770820410609745804738512, 14.41372710534778237529570306286, 14.82519161386866466903445628252, 16.14328077297247309570913225706, 17.677609167586493772292410566528, 18.74002943382334547807089331579, 19.747396761203559272346685787673, 20.336408262379719620779062426133, 21.36594839033843814111954450315, 22.50188100774064079907584289439, 23.407713320577184524710375100628, 24.339069125851163806557174352937, 25.008596044475990158492213022975, 26.01772320082470667580256714866, 27.22251719966340627763621183128