| L(s) = 1 | − 4·4-s + 6·5-s − 2·9-s + 12·16-s − 24·20-s + 17·25-s − 8·31-s + 8·36-s + 4·37-s − 6·41-s − 2·43-s − 12·45-s − 13·49-s − 12·59-s − 2·61-s − 32·64-s − 14·73-s + 72·80-s − 5·81-s + 24·83-s − 68·100-s + 28·103-s − 36·107-s + 12·113-s − 13·121-s + 32·124-s + 18·125-s + ⋯ |
| L(s) = 1 | − 2·4-s + 2.68·5-s − 2/3·9-s + 3·16-s − 5.36·20-s + 17/5·25-s − 1.43·31-s + 4/3·36-s + 0.657·37-s − 0.937·41-s − 0.304·43-s − 1.78·45-s − 1.85·49-s − 1.56·59-s − 0.256·61-s − 4·64-s − 1.63·73-s + 8.04·80-s − 5/9·81-s + 2.63·83-s − 6.79·100-s + 2.75·103-s − 3.48·107-s + 1.12·113-s − 1.18·121-s + 2.87·124-s + 1.60·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 606841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 606841 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 41 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.296499449630110216369102611507, −7.923460562946246199886108427469, −7.39532327685859741640586695460, −6.39084306102520193435942181609, −6.30092624798927176107946148012, −5.75116326140982635046813181964, −5.38402383492515633625005346052, −5.03912355415234199771433602492, −4.66113670445354521481725901186, −3.85615252208514256878324668324, −3.28403235558794973310569727230, −2.67183638730115156824697116428, −1.79554652871898727064930174155, −1.35486074388197716928380982194, 0,
1.35486074388197716928380982194, 1.79554652871898727064930174155, 2.67183638730115156824697116428, 3.28403235558794973310569727230, 3.85615252208514256878324668324, 4.66113670445354521481725901186, 5.03912355415234199771433602492, 5.38402383492515633625005346052, 5.75116326140982635046813181964, 6.30092624798927176107946148012, 6.39084306102520193435942181609, 7.39532327685859741640586695460, 7.923460562946246199886108427469, 8.296499449630110216369102611507
