| L(s) = 1 | + 2-s − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s
+ 10-s + 11-s − 3·12-s + 14-s − 3·15-s + 16-s − 3·17-s + 6·18-s
− 19-s + 20-s − 3·21-s + 22-s − 4·23-s − 3·24-s + 25-s − 9·27-s
+ 28-s − 5·29-s − 3·30-s + ⋯
|
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s
+ 0.316·10-s + 0.301·11-s − 0.866·12-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s
− 0.229·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 1.73·27-s
+ 0.188·28-s − 0.928·29-s − 0.547·30-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & -\, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & -\, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;5,\;11,\;43\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11,\;43\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
| show more | |
| show less | |
\[\begin{aligned}
L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Imaginary part of the first few zeros on the critical line
−17.84675313983433, −17.37144243692654, −17.12728629931520, −16.17314131172689, −15.95467970084369, −15.16654536832013, −14.41239237572319, −13.81392100269725, −13.00048065709477, −12.57794938161210, −11.94069544365215, −11.29917364657803, −10.94953670876546, −10.25506790952758, −9.592626527551025, −8.622665179781464, −7.613742688823653, −6.747634030335561, −6.464231475252572, −5.465713684870454, −5.306018404096157, −4.354115097724995, −3.718242750676210, −2.207203146962727, −1.413343267926647, 0,
1.413343267926647, 2.207203146962727, 3.718242750676210, 4.354115097724995, 5.306018404096157, 5.465713684870454, 6.464231475252572, 6.747634030335561, 7.613742688823653, 8.622665179781464, 9.592626527551025, 10.25506790952758, 10.94953670876546, 11.29917364657803, 11.94069544365215, 12.57794938161210, 13.00048065709477, 13.81392100269725, 14.41239237572319, 15.16654536832013, 15.95467970084369, 16.17314131172689, 17.12728629931520, 17.37144243692654, 17.84675313983433
