| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 6·5-s + 6·6-s − 16·7-s − 8·8-s + 9·9-s
− 12·10-s + 12·11-s − 12·12-s + 38·13-s + 32·14-s − 18·15-s + 16·16-s − 126·17-s
− 18·18-s + 20·19-s + 24·20-s + 48·21-s − 24·22-s + 168·23-s + 24·24-s − 89·25-s
− 76·26-s − 27·27-s − 64·28-s + ⋯
|
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.536·5-s + 0.408·6-s − 0.863·7-s − 0.353·8-s + 1/3·9-s
− 0.379·10-s + 0.328·11-s − 0.288·12-s + 0.810·13-s + 0.610·14-s − 0.309·15-s + 1/4·16-s − 1.79·17-s
− 0.235·18-s + 0.241·19-s + 0.268·20-s + 0.498·21-s − 0.232·22-s + 1.52·23-s + 0.204·24-s − 0.711·25-s
− 0.573·26-s − 0.192·27-s − 0.431·28-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(4-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
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 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 126 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 - 168 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 88 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 42 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 + 96 T + p^{3} T^{2} \) |
| 53 | \( 1 - 198 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 538 T + p^{3} T^{2} \) |
| 67 | \( 1 - 884 T + p^{3} T^{2} \) |
| 71 | \( 1 - 792 T + p^{3} T^{2} \) |
| 73 | \( 1 - 218 T + p^{3} T^{2} \) |
| 79 | \( 1 + 520 T + p^{3} T^{2} \) |
| 83 | \( 1 + 492 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1154 T + p^{3} T^{2} \) |
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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
−22.97901126778615505977980355311, −21.58778001713410487232827259598, −19.86669072742657907925171947529, −18.33094748384698500881885833386, −17.06455999699601621437849501998, −15.67334368036257763172445032023, −13.15736614711876148220321478103, −11.07166341738426389307147212093, −9.298914371498593683037799278765, −6.48044763846723198372779155495,
6.48044763846723198372779155495, 9.298914371498593683037799278765, 11.07166341738426389307147212093, 13.15736614711876148220321478103, 15.67334368036257763172445032023, 17.06455999699601621437849501998, 18.33094748384698500881885833386, 19.86669072742657907925171947529, 21.58778001713410487232827259598, 22.97901126778615505977980355311
