| L(s) = 1 | − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s − 11-s − 13-s
+ 4·14-s + 16-s − 17-s + 7·19-s − 20-s + 22-s + 4·23-s + 25-s
+ 26-s − 4·28-s + 29-s − 9·31-s − 32-s + 34-s + 4·35-s + 4·37-s
− 7·38-s + 40-s + 8·41-s + ⋯
|
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.277·13-s
+ 1.06·14-s + 1/4·16-s − 0.242·17-s + 1.60·19-s − 0.223·20-s + 0.213·22-s + 0.834·23-s + 1/5·25-s
+ 0.196·26-s − 0.755·28-s + 0.185·29-s − 1.61·31-s − 0.176·32-s + 0.171·34-s + 0.676·35-s + 0.657·37-s
− 1.13·38-s + 0.158·40-s + 1.24·41-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & \, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{2,\;3,\;5,\;11,\;17\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 17 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.89329284932846, −15.77431370436769, −14.77424383707840, −14.49624941083366, −13.49105054778978, −13.02928180272976, −12.64551332297067, −11.83083799552430, −11.50866626510747, −10.67039281555637, −10.24329885196541, −9.580735291658147, −9.115965733717806, −8.729191956029319, −7.639882384929231, −7.308228435870612, −6.924974893367380, −5.869979828880284, −5.659062779657923, −4.553484812571371, −3.760208710929760, −2.985261547291284, −2.657050448484445, −1.341406160318551, −0.4253817373562672,
0.4253817373562672, 1.341406160318551, 2.657050448484445, 2.985261547291284, 3.760208710929760, 4.553484812571371, 5.659062779657923, 5.869979828880284, 6.924974893367380, 7.308228435870612, 7.639882384929231, 8.729191956029319, 9.115965733717806, 9.580735291658147, 10.24329885196541, 10.67039281555637, 11.50866626510747, 11.83083799552430, 12.64551332297067, 13.02928180272976, 13.49105054778978, 14.49624941083366, 14.77424383707840, 15.77431370436769, 15.89329284932846
