| L(s) = 1 | − 3·4-s − 2·7-s + 9-s + 8·11-s − 4·13-s + 5·16-s − 12·17-s − 6·25-s + 6·28-s − 4·29-s − 3·36-s + 12·37-s − 8·43-s − 24·44-s + 3·49-s + 12·52-s + 6·53-s + 24·59-s − 2·63-s − 3·64-s + 36·68-s − 16·77-s + 81-s − 28·89-s + 8·91-s + 36·97-s + 8·99-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 0.755·7-s + 1/3·9-s + 2.41·11-s − 1.10·13-s + 5/4·16-s − 2.91·17-s − 6/5·25-s + 1.13·28-s − 0.742·29-s − 1/2·36-s + 1.97·37-s − 1.21·43-s − 3.61·44-s + 3/7·49-s + 1.66·52-s + 0.824·53-s + 3.12·59-s − 0.251·63-s − 3/8·64-s + 4.36·68-s − 1.82·77-s + 1/9·81-s − 2.96·89-s + 0.838·91-s + 3.65·97-s + 0.804·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1238769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1238769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 53 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−7.81047435156176663326713569748, −7.25047783802838427330028450541, −6.83203797015686736081127611784, −6.51635748516493376245552793712, −6.20118360518008255279238849027, −5.48527913616952883926237154637, −5.04724637002800076836385273071, −4.25883364417471195561731212514, −4.13559084050773741974089362056, −4.08188707091001456471412468721, −3.26858741332919914473868805629, −2.34403251626970777143753618919, −1.92169220994903191685155924585, −0.836505510790425927527221533043, 0,
0.836505510790425927527221533043, 1.92169220994903191685155924585, 2.34403251626970777143753618919, 3.26858741332919914473868805629, 4.08188707091001456471412468721, 4.13559084050773741974089362056, 4.25883364417471195561731212514, 5.04724637002800076836385273071, 5.48527913616952883926237154637, 6.20118360518008255279238849027, 6.51635748516493376245552793712, 6.83203797015686736081127611784, 7.25047783802838427330028450541, 7.81047435156176663326713569748
