| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s + 3·9-s − 2·10-s − 4·11-s − 13-s + 16-s − 6·17-s + 3·18-s + 12·19-s − 2·20-s − 4·22-s − 8·23-s − 7·25-s − 26-s + 32-s − 6·34-s + 3·36-s + 6·37-s + 12·38-s − 2·40-s − 4·44-s − 6·45-s − 8·46-s − 13·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s + 9-s − 0.632·10-s − 1.20·11-s − 0.277·13-s + 1/4·16-s − 1.45·17-s + 0.707·18-s + 2.75·19-s − 0.447·20-s − 0.852·22-s − 1.66·23-s − 7/5·25-s − 0.196·26-s + 0.176·32-s − 1.02·34-s + 1/2·36-s + 0.986·37-s + 1.94·38-s − 0.316·40-s − 0.603·44-s − 0.894·45-s − 1.17·46-s − 1.85·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;13\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 3.
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 13 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]
Imaginary part of the first few zeros on the critical line
−8.372504302205780920168328513074, −7.952773675345803042836433200054, −7.54927518755809077497495838664, −7.45985560667588965978165906144, −6.83793489243240202981608052708, −5.99269128295739404934949048165, −5.88908450915949561478136129218, −5.03252986183832812949281939535, −4.61153453491182141362175201622, −4.25770102664182839693622055004, −3.54070244821137046528273748949, −3.09063283442029069651336655846, −2.27841740887741805213486299325, −1.52941733184663708479315761299, 0,
1.52941733184663708479315761299, 2.27841740887741805213486299325, 3.09063283442029069651336655846, 3.54070244821137046528273748949, 4.25770102664182839693622055004, 4.61153453491182141362175201622, 5.03252986183832812949281939535, 5.88908450915949561478136129218, 5.99269128295739404934949048165, 6.83793489243240202981608052708, 7.45985560667588965978165906144, 7.54927518755809077497495838664, 7.952773675345803042836433200054, 8.372504302205780920168328513074
