L(s) = 1 | + (−0.248 + 0.968i)2-s + (0.904 − 0.425i)3-s + (−0.876 − 0.481i)4-s + (0.187 + 0.982i)6-s + (−0.951 + 0.309i)7-s + (0.684 − 0.728i)8-s + (0.637 − 0.770i)9-s + (−0.998 − 0.0627i)12-s + (0.770 + 0.637i)13-s + (−0.0627 − 0.998i)14-s + (0.535 + 0.844i)16-s + (−0.904 − 0.425i)17-s + (0.587 + 0.809i)18-s + (−0.187 − 0.982i)19-s + (−0.728 + 0.684i)21-s + ⋯ |
L(s) = 1 | + (−0.248 + 0.968i)2-s + (0.904 − 0.425i)3-s + (−0.876 − 0.481i)4-s + (0.187 + 0.982i)6-s + (−0.951 + 0.309i)7-s + (0.684 − 0.728i)8-s + (0.637 − 0.770i)9-s + (−0.998 − 0.0627i)12-s + (0.770 + 0.637i)13-s + (−0.0627 − 0.998i)14-s + (0.535 + 0.844i)16-s + (−0.904 − 0.425i)17-s + (0.587 + 0.809i)18-s + (−0.187 − 0.982i)19-s + (−0.728 + 0.684i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.517837312 + 0.2379541596i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.517837312 + 0.2379541596i\) |
\(L(1)\) |
\(\approx\) |
\(1.079004793 + 0.2611045943i\) |
\(L(1)\) |
\(\approx\) |
\(1.079004793 + 0.2611045943i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.248 + 0.968i)T \) |
| 3 | \( 1 + (0.904 - 0.425i)T \) |
| 7 | \( 1 + (-0.951 + 0.309i)T \) |
| 13 | \( 1 + (0.770 + 0.637i)T \) |
| 17 | \( 1 + (-0.904 - 0.425i)T \) |
| 19 | \( 1 + (-0.187 - 0.982i)T \) |
| 23 | \( 1 + (0.844 + 0.535i)T \) |
| 29 | \( 1 + (-0.992 + 0.125i)T \) |
| 31 | \( 1 + (0.728 + 0.684i)T \) |
| 37 | \( 1 + (0.770 + 0.637i)T \) |
| 41 | \( 1 + (0.637 - 0.770i)T \) |
| 43 | \( 1 + (-0.951 - 0.309i)T \) |
| 47 | \( 1 + (0.982 + 0.187i)T \) |
| 53 | \( 1 + (-0.982 - 0.187i)T \) |
| 59 | \( 1 + (0.929 + 0.368i)T \) |
| 61 | \( 1 + (0.929 - 0.368i)T \) |
| 67 | \( 1 + (0.904 + 0.425i)T \) |
| 71 | \( 1 + (-0.425 - 0.904i)T \) |
| 73 | \( 1 + (0.248 - 0.968i)T \) |
| 79 | \( 1 + (0.728 - 0.684i)T \) |
| 83 | \( 1 + (0.684 - 0.728i)T \) |
| 89 | \( 1 + (-0.0627 - 0.998i)T \) |
| 97 | \( 1 + (0.481 - 0.876i)T \) |
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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
−20.47199926891279738880913488560, −20.29981430855174165368431083896, −19.29506539907257602055105792683, −18.90085143173438154509506559411, −18.064516966240683406656901083228, −16.978945785478182588357202870624, −16.34867097355128988209110468189, −15.422194735611007510041957000868, −14.61847488607679783925619495273, −13.68063804720127813860993336207, −12.998846113651742948428965258241, −12.69968482888166692128621702065, −11.252920422060871657994138584966, −10.669197076946834279395218864495, −9.8778922382210857228035796186, −9.321006240526871937674007006220, −8.43581221288603509864052053622, −7.86843635182794885014440541481, −6.69217964004160853471937025117, −5.52469672580711735209701602433, −4.23821493748357940411994482664, −3.79975095160601392999299666254, −2.9288058064389781947272120613, −2.16956516383783353572033553938, −0.95107872252985878680475309941,
0.74295124646680660838147577774, 2.00627547072671815523297576096, 3.12339668662398742746510628523, 3.99207851606982883763717004422, 4.98822663961965886089152969549, 6.21128954918924030587995204218, 6.75231549726243802370912131037, 7.37547903108524638776123661448, 8.469690650856592315654002424031, 9.12219768676685503416464894678, 9.43152433262320675291585785731, 10.59484434914373388224268098906, 11.72774646915273325694535013964, 13.027494957128013717211955637146, 13.24961443196894581731347243448, 13.99760056389046002484142699995, 14.94154942887184445770795931095, 15.606155153872234065673475470611, 16.05196358319123375022805689663, 17.10872012812904752406997298709, 17.9108045236229840829456493234, 18.69359637672352548764545486013, 19.17736730877091109631717720418, 19.83907772251919145768579577056, 20.76472342639111438608347782338