| L(s) = 1 | − 2·2-s − 4-s + 7-s + 8·8-s − 2·9-s − 10·11-s − 2·14-s − 7·16-s + 4·18-s + 20·22-s − 18·23-s − 25-s − 28-s − 12·29-s − 14·32-s + 2·36-s + 16·37-s − 16·43-s + 10·44-s + 36·46-s − 6·49-s + 2·50-s + 12·53-s + 8·56-s + 24·58-s − 2·63-s + 35·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 0.377·7-s + 2.82·8-s − 2/3·9-s − 3.01·11-s − 0.534·14-s − 7/4·16-s + 0.942·18-s + 4.26·22-s − 3.75·23-s − 1/5·25-s − 0.188·28-s − 2.22·29-s − 2.47·32-s + 1/3·36-s + 2.63·37-s − 2.43·43-s + 1.50·44-s + 5.30·46-s − 6/7·49-s + 0.282·50-s + 1.64·53-s + 1.06·56-s + 3.15·58-s − 0.251·63-s + 35/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.311539887357104546707428260170, −8.287570693733701613432166498970, −7.981017946213055065368454386439, −7.66039763333771123227596439320, −7.21457739085084948584915045817, −5.98053927859956525814380109628, −5.60157612088078313975836309422, −5.33453746338054689929021929846, −4.44670425649641091969326941871, −4.25666915587159511841100490901, −3.30805769426848244281074322907, −2.30317115007787891842697175045, −1.79788073888911014217168687538, 0, 0,
1.79788073888911014217168687538, 2.30317115007787891842697175045, 3.30805769426848244281074322907, 4.25666915587159511841100490901, 4.44670425649641091969326941871, 5.33453746338054689929021929846, 5.60157612088078313975836309422, 5.98053927859956525814380109628, 7.21457739085084948584915045817, 7.66039763333771123227596439320, 7.981017946213055065368454386439, 8.287570693733701613432166498970, 8.311539887357104546707428260170
