| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 11-s + 2·13-s
− 14-s + 16-s − 6·17-s − 4·19-s + 20-s + 22-s − 6·23-s + 25-s
− 2·26-s + 28-s + 6·29-s + 2·31-s − 32-s + 6·34-s + 35-s + 2·37-s
+ 4·38-s − 40-s + 12·41-s + ⋯
|
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.554·13-s
− 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s
− 0.392·26-s + 0.188·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.328·37-s
+ 0.648·38-s − 0.158·40-s + 1.87·41-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & -\, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & -\, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−17.59862415020751, −17.07982086181961, −16.33707120095797, −15.73211044616070, −15.41890302965346, −14.50267185967208, −14.02764960946052, −13.33639705738793, −12.71775331496483, −12.08455228572673, −11.21796799055397, −10.86223997168004, −10.27919446176191, −9.526426979904304, −8.937378818078003, −8.295710805212396, −7.836168708492288, −6.879395609361139, −6.252735576415554, −5.786824933354323, −4.602181488446775, −4.143296596633825, −2.808969531356003, −2.202130005773945, −1.296944977976799, 0,
1.296944977976799, 2.202130005773945, 2.808969531356003, 4.143296596633825, 4.602181488446775, 5.786824933354323, 6.252735576415554, 6.879395609361139, 7.836168708492288, 8.295710805212396, 8.937378818078003, 9.526426979904304, 10.27919446176191, 10.86223997168004, 11.21796799055397, 12.08455228572673, 12.71775331496483, 13.33639705738793, 14.02764960946052, 14.50267185967208, 15.41890302965346, 15.73211044616070, 16.33707120095797, 17.07982086181961, 17.59862415020751
