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
−20.76472342639111438608347782338, −19.83907772251919145768579577056, −19.17736730877091109631717720418, −18.69359637672352548764545486013, −17.9108045236229840829456493234, −17.10872012812904752406997298709, −16.05196358319123375022805689663, −15.606155153872234065673475470611, −14.94154942887184445770795931095, −13.99760056389046002484142699995, −13.24961443196894581731347243448, −13.027494957128013717211955637146, −11.72774646915273325694535013964, −10.59484434914373388224268098906, −9.43152433262320675291585785731, −9.12219768676685503416464894678, −8.469690650856592315654002424031, −7.37547903108524638776123661448, −6.75231549726243802370912131037, −6.21128954918924030587995204218, −4.98822663961965886089152969549, −3.99207851606982883763717004422, −3.12339668662398742746510628523, −2.00627547072671815523297576096, −0.74295124646680660838147577774,
0.95107872252985878680475309941, 2.16956516383783353572033553938, 2.9288058064389781947272120613, 3.79975095160601392999299666254, 4.23821493748357940411994482664, 5.52469672580711735209701602433, 6.69217964004160853471937025117, 7.86843635182794885014440541481, 8.43581221288603509864052053622, 9.321006240526871937674007006220, 9.8778922382210857228035796186, 10.669197076946834279395218864495, 11.252920422060871657994138584966, 12.69968482888166692128621702065, 12.998846113651742948428965258241, 13.68063804720127813860993336207, 14.61847488607679783925619495273, 15.422194735611007510041957000868, 16.34867097355128988209110468189, 16.978945785478182588357202870624, 18.064516966240683406656901083228, 18.90085143173438154509506559411, 19.29506539907257602055105792683, 20.29981430855174165368431083896, 20.47199926891279738880913488560