| L(s) = 1 | + 1.67·2-s + 3-s + 0.795·4-s + 2.68·5-s + 1.67·6-s − 2.01·8-s + 9-s + 4.48·10-s
− 5.74·11-s + 0.795·12-s − 4.92·13-s + 2.68·15-s − 4.95·16-s − 2.19·17-s + 1.67·18-s + 3.26·19-s
+ 2.13·20-s − 9.61·22-s − 6.02·23-s − 2.01·24-s + 2.19·25-s − 8.23·26-s + 27-s − 4.01·29-s
+ 4.48·30-s + 7.54·31-s − 4.26·32-s + ⋯
|
| L(s) = 1 | + 1.18·2-s + 0.577·3-s + 0.397·4-s + 1.19·5-s + 0.682·6-s − 0.712·8-s + 0.333·9-s + 1.41·10-s
− 1.73·11-s + 0.229·12-s − 1.36·13-s + 0.692·15-s − 1.23·16-s − 0.531·17-s + 0.394·18-s + 0.748·19-s
+ 0.477·20-s − 2.04·22-s − 1.25·23-s − 0.411·24-s + 0.439·25-s − 1.61·26-s + 0.192·27-s − 0.744·29-s
+ 0.819·30-s + 1.35·31-s − 0.753·32-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & -\, \Lambda(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & -\, \Lambda(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{3,\;7,\;41\}$,
\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 5 | \( 1 - 2.68T + 5T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 13 | \( 1 + 4.92T + 13T^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 + 6.02T + 23T^{2} \) |
| 29 | \( 1 + 4.01T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 43 | \( 1 + 4.98T + 43T^{2} \) |
| 47 | \( 1 + 4.57T + 47T^{2} \) |
| 53 | \( 1 - 1.25T + 53T^{2} \) |
| 59 | \( 1 + 6.36T + 59T^{2} \) |
| 61 | \( 1 + 6.05T + 61T^{2} \) |
| 67 | \( 1 - 1.90T + 67T^{2} \) |
| 71 | \( 1 + 9.71T + 71T^{2} \) |
| 73 | \( 1 - 6.59T + 73T^{2} \) |
| 79 | \( 1 - 3.22T + 79T^{2} \) |
| 83 | \( 1 + 9.82T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 7.28T + 97T^{2} \) |
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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
−7.73523068252753218272464144334, −6.85355626196598751749644444732, −6.01132616391271459380083912681, −5.46573150348553535912210344882, −4.85795821201146021626440623222, −4.23972017825933061143878101532, −2.99140254605097366197796000491, −2.62425712259383727581337264029, −1.90121731472281249403182227332, 0,
1.90121731472281249403182227332, 2.62425712259383727581337264029, 2.99140254605097366197796000491, 4.23972017825933061143878101532, 4.85795821201146021626440623222, 5.46573150348553535912210344882, 6.01132616391271459380083912681, 6.85355626196598751749644444732, 7.73523068252753218272464144334
