| L(s) = 1 | + (−0.336 + 1.90i)2-s + (1.43 + 0.966i)3-s + (−1.64 − 0.600i)4-s
+ (−1.09 − 3.00i)5-s + (−2.32 + 2.41i)6-s + (−1.23 − 2.13i)7-s
+ (−0.237 + 0.412i)8-s + (1.13 + 2.77i)9-s + (6.11 − 1.07i)10-s
+ (−2.11 − 1.21i)11-s + (−1.78 − 2.45i)12-s + (1.65 + 1.97i)13-s
+ (4.48 − 1.63i)14-s + (1.33 − 5.38i)15-s + (−3.39 − 2.84i)16-s
+ (3.75 + 0.662i)17-s + ⋯
|
| L(s) = 1 | + (−0.237 + 1.34i)2-s + (0.829 + 0.557i)3-s + (−0.824 − 0.300i)4-s
+ (−0.489 − 1.34i)5-s + (−0.950 + 0.987i)6-s + (−0.465 − 0.805i)7-s
+ (−0.0841 + 0.145i)8-s + (0.377 + 0.926i)9-s + (1.93 − 0.340i)10-s
+ (−0.636 − 0.367i)11-s + (−0.516 − 0.708i)12-s + (0.458 + 0.547i)13-s
+ (1.19 − 0.435i)14-s + (0.344 − 1.39i)15-s + (−0.848 − 0.712i)16-s
+ (0.911 + 0.160i)17-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & (0.0300 - 0.999i)\, \overline{\Lambda}(2-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr
=\mathstrut & (0.0300 - 0.999i)\, \overline{\Lambda}(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{3,\;19\}$,
\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.43 - 0.966i)T \) |
| 19 | \( 1 + (2.59 - 3.50i)T \) |
| good | 2 | \( 1 + (0.336 - 1.90i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (1.09 + 3.00i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.23 + 2.13i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.11 + 1.21i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.65 - 1.97i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.75 - 0.662i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.817 + 2.24i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.705 - 3.99i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.82 - 2.20i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 9.08iT - 37T^{2} \) |
| 41 | \( 1 + (4.39 + 3.68i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.86 + 1.40i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-3.04 + 0.536i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.11 - 1.13i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.78 - 10.1i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.932 - 0.339i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (6.74 - 1.18i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.208 - 0.0757i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-12.8 - 10.8i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.67 + 3.18i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (2.45 - 1.41i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.53 + 5.48i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-3.82 - 0.673i)T + (91.1 + 33.1i)T^{2} \) |
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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
−15.86205764318458976216103838745, −14.62687094278187924858535298065, −13.67562729205643419634523263233, −12.50746277755758118646665997586, −10.55132057614183621244177802120, −9.056572118784488988334963947531, −8.342886032887341889969256522064, −7.32504642368223883126646029561, −5.43594230161091345774193366818, −3.99407877930334346704347257383,
2.54586146430352091830856281159, 3.38830114165394806824490771880, 6.51189619201459778390899520089, 7.892575599594483906500224674935, 9.370464394386344758473488962986, 10.41909379196453081170428931386, 11.55401989751276207136648704957, 12.55881422783037588777666512152, 13.51553875560096729367752359799, 15.05425671955109804187837604095
