L(s) = 1 | + 4-s − 2·5-s − 4·7-s − 6·9-s − 3·16-s − 4·17-s − 2·20-s + 12·23-s + 3·25-s − 4·28-s + 8·35-s − 6·36-s − 12·43-s + 12·45-s − 4·47-s − 2·49-s − 20·61-s + 24·63-s − 7·64-s − 4·68-s + 28·73-s + 6·80-s + 27·81-s + 28·83-s + 8·85-s + 12·92-s + 3·100-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 0.894·5-s − 1.51·7-s − 2·9-s − 3/4·16-s − 0.970·17-s − 0.447·20-s + 2.50·23-s + 3/5·25-s − 0.755·28-s + 1.35·35-s − 36-s − 1.82·43-s + 1.78·45-s − 0.583·47-s − 2/7·49-s − 2.56·61-s + 3.02·63-s − 7/8·64-s − 0.485·68-s + 3.27·73-s + 0.670·80-s + 3·81-s + 3.07·83-s + 0.867·85-s + 1.25·92-s + 3/10·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
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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
−9.014339022682654525788748670100, −8.897749114707035393122973926869, −8.268810609789432089132763144540, −7.900675296222237214344109636738, −7.59114460523762221657093265967, −6.84033795625724349746851931118, −6.60416628536008257449352409647, −6.47890268486684764787876703445, −6.12713122160770754987246398695, −5.27056767614506292425312527395, −4.96142906670360965911811463386, −4.79290576208334763260373758918, −3.76543315121891478392405415145, −3.48293467799567444191004825750, −3.10867401144006853271230976319, −2.65466888321512335124647307280, −2.31254564290464559679453706839, −1.20098488214232486959183582723, 0, 0,
1.20098488214232486959183582723, 2.31254564290464559679453706839, 2.65466888321512335124647307280, 3.10867401144006853271230976319, 3.48293467799567444191004825750, 3.76543315121891478392405415145, 4.79290576208334763260373758918, 4.96142906670360965911811463386, 5.27056767614506292425312527395, 6.12713122160770754987246398695, 6.47890268486684764787876703445, 6.60416628536008257449352409647, 6.84033795625724349746851931118, 7.59114460523762221657093265967, 7.900675296222237214344109636738, 8.268810609789432089132763144540, 8.897749114707035393122973926869, 9.014339022682654525788748670100