| L(s) = 1 | + 3·5-s − 3·7-s − 2·11-s + 2·17-s − 3·19-s − 2·23-s + 4·25-s − 6·31-s − 9·35-s − 4·37-s + 10·41-s − 2·43-s + 3·47-s + 2·49-s + 3·53-s − 6·55-s + 12·59-s + 61-s + 10·67-s − 10·71-s + 7·73-s + 6·77-s − 8·79-s − 6·83-s + 6·85-s − 6·89-s − 9·95-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s − 0.603·11-s + 0.485·17-s − 0.688·19-s − 0.417·23-s + 4/5·25-s − 1.07·31-s − 1.52·35-s − 0.657·37-s + 1.56·41-s − 0.304·43-s + 0.437·47-s + 2/7·49-s + 0.412·53-s − 0.809·55-s + 1.56·59-s + 0.128·61-s + 1.22·67-s − 1.18·71-s + 0.819·73-s + 0.683·77-s − 0.900·79-s − 0.658·83-s + 0.650·85-s − 0.635·89-s − 0.923·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 463 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.07690366742594, −12.96641497326040, −12.58525418784040, −12.04909496472024, −11.27898696465144, −10.92352475681986, −10.19198117599158, −10.05007549420762, −9.706270468730280, −9.019224798074399, −8.787895430255686, −8.091646971545882, −7.463712082343126, −6.990323749772053, −6.466901546588543, −6.015280763986192, −5.530637210324306, −5.305074785151857, −4.417691192691124, −3.848748369929503, −3.282804386337290, −2.620727830194970, −2.232575772494936, −1.633361093748897, −0.7792082365212458, 0,
0.7792082365212458, 1.633361093748897, 2.232575772494936, 2.620727830194970, 3.282804386337290, 3.848748369929503, 4.417691192691124, 5.305074785151857, 5.530637210324306, 6.015280763986192, 6.466901546588543, 6.990323749772053, 7.463712082343126, 8.091646971545882, 8.787895430255686, 9.019224798074399, 9.706270468730280, 10.05007549420762, 10.19198117599158, 10.92352475681986, 11.27898696465144, 12.04909496472024, 12.58525418784040, 12.96641497326040, 13.07690366742594
