| L(s) = 1 | + 27·3-s + 110·5-s + 504·7-s + 729·9-s + 3.81e3·11-s + 9.57e3·13-s + 2.97e3·15-s + 2.60e4·17-s
− 3.83e4·19-s + 1.36e4·21-s − 7.11e4·23-s − 6.60e4·25-s + 1.96e4·27-s + 7.42e4·29-s − 2.75e5·31-s + 1.02e5·33-s
+ 5.54e4·35-s − 2.66e5·37-s + 2.58e5·39-s + 6.84e5·41-s + 2.45e5·43-s + 8.01e4·45-s + 4.78e5·47-s − 5.69e5·49-s
+ 7.04e5·51-s − 5.69e5·53-s + 4.19e5·55-s + ⋯
|
| L(s) = 1 | + 0.577·3-s + 0.393·5-s + 0.555·7-s + 1/3·9-s + 0.863·11-s + 1.20·13-s + 0.227·15-s + 1.28·17-s
− 1.28·19-s + 0.320·21-s − 1.21·23-s − 0.845·25-s + 0.192·27-s + 0.565·29-s − 1.66·31-s + 0.498·33-s
+ 0.218·35-s − 0.865·37-s + 0.697·39-s + 1.55·41-s + 0.471·43-s + 0.131·45-s + 0.672·47-s − 0.691·49-s
+ 0.743·51-s − 0.525·53-s + 0.339·55-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(8-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;3\}$,
\(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 - 22 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 72 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 3812 T + p^{7} T^{2} \) |
| 13 | \( 1 - 9574 T + p^{7} T^{2} \) |
| 17 | \( 1 - 26098 T + p^{7} T^{2} \) |
| 19 | \( 1 + 38308 T + p^{7} T^{2} \) |
| 23 | \( 1 + 71128 T + p^{7} T^{2} \) |
| 29 | \( 1 - 74262 T + p^{7} T^{2} \) |
| 31 | \( 1 + 275680 T + p^{7} T^{2} \) |
| 37 | \( 1 + 266610 T + p^{7} T^{2} \) |
| 41 | \( 1 - 684762 T + p^{7} T^{2} \) |
| 43 | \( 1 - 245956 T + p^{7} T^{2} \) |
| 47 | \( 1 - 478800 T + p^{7} T^{2} \) |
| 53 | \( 1 + 569410 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1525324 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2640458 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1416236 T + p^{7} T^{2} \) |
| 71 | \( 1 + 3511304 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4738618 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4661488 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5729252 T + p^{7} T^{2} \) |
| 89 | \( 1 - 11993514 T + p^{7} T^{2} \) |
| 97 | \( 1 - 7150754 T + p^{7} T^{2} \) |
| 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
−16.09190602319062579992194893959, −14.63221200023629279949614909928, −13.83671489129554663852074268933, −12.33578416001402223357669799575, −10.76228868234596434615313168621, −9.243042176868312796418724021022, −7.953790694673518715589943890035, −6.05317916615898200552832841953, −3.88377140984588093885913230902, −1.66142724699780075595490606581,
1.66142724699780075595490606581, 3.88377140984588093885913230902, 6.05317916615898200552832841953, 7.953790694673518715589943890035, 9.243042176868312796418724021022, 10.76228868234596434615313168621, 12.33578416001402223357669799575, 13.83671489129554663852074268933, 14.63221200023629279949614909928, 16.09190602319062579992194893959
