| L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s − 4·5-s + 6·6-s − 4·7-s + 6·9-s + 8·10-s
− 6·11-s − 6·12-s − 4·13-s + 8·14-s + 12·15-s − 4·16-s − 4·17-s − 12·18-s
− 7·19-s − 8·20-s + 12·21-s + 12·22-s − 6·23-s + 11·25-s + 8·26-s − 9·27-s
− 8·28-s − 6·29-s − 24·30-s + ⋯
|
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s − 1.78·5-s + 2.44·6-s − 1.51·7-s + 2·9-s + 2.52·10-s
− 1.80·11-s − 1.73·12-s − 1.10·13-s + 2.13·14-s + 3.09·15-s − 16-s − 0.970·17-s − 2.82·18-s
− 1.60·19-s − 1.78·20-s + 2.61·21-s + 2.55·22-s − 1.25·23-s + 11/5·25-s + 1.56·26-s − 1.73·27-s
− 1.51·28-s − 1.11·29-s − 4.38·30-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & -\, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & -\, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \neq 5077$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 5077$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 5077 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| show more | |
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\[\begin{aligned}
L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Imaginary part of the first few zeros on the critical line
−18.55973951718974, −18.07306090799613, −17.41152136149437, −16.64312940081154, −16.44048490106365, −15.89479293723709, −15.51534707653608, −14.94156032954847, −13.15163660315273, −12.97272258207286, −12.28722890382493, −11.68694809088531, −11.03344123514270, −10.49585360108396, −10.20346324266066, −9.382178911171940, −8.476801942623500, −7.706794648113253, −7.342814979539648, −6.622504613407707, −6.011922752986395, −4.754431515963406, −4.470551513310098, −3.262443555978757, −2.052472858479940, 0, 0, 0,
2.052472858479940, 3.262443555978757, 4.470551513310098, 4.754431515963406, 6.011922752986395, 6.622504613407707, 7.342814979539648, 7.706794648113253, 8.476801942623500, 9.382178911171940, 10.20346324266066, 10.49585360108396, 11.03344123514270, 11.68694809088531, 12.28722890382493, 12.97272258207286, 13.15163660315273, 14.94156032954847, 15.51534707653608, 15.89479293723709, 16.44048490106365, 16.64312940081154, 17.41152136149437, 18.07306090799613, 18.55973951718974
