| L(s) = 1 | − 3·4-s + 4·7-s − 3·9-s + 5·16-s + 6·19-s − 6·25-s − 12·28-s + 9·36-s + 12·43-s − 2·49-s + 12·61-s − 12·63-s − 3·64-s − 4·73-s − 18·76-s + 9·81-s + 18·100-s + 20·112-s − 121-s + 127-s + 131-s + 24·133-s + 137-s + 139-s − 15·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1.51·7-s − 9-s + 5/4·16-s + 1.37·19-s − 6/5·25-s − 2.26·28-s + 3/2·36-s + 1.82·43-s − 2/7·49-s + 1.53·61-s − 1.51·63-s − 3/8·64-s − 0.468·73-s − 2.06·76-s + 81-s + 9/5·100-s + 1.88·112-s − 0.0909·121-s + 0.0887·127-s + 0.0873·131-s + 2.08·133-s + 0.0854·137-s + 0.0848·139-s − 5/4·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.219338914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.219338914\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| 19 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.657345504637790262473996328474, −8.130186670439284459309177922112, −7.972151940820601836697648015290, −7.53997736896322143284529825653, −6.92329970739893381897680957631, −6.07128255175903051808969963153, −5.67207155470213255326225150387, −5.21166518215897327046332845629, −4.94437307110149956745580470362, −4.30040508629908295283598427180, −3.88860327572794518458625049254, −3.22754362893966258462003953821, −2.45452637871016402228825655496, −1.58724456551048532456897050947, −0.65533493528900419546138704462,
0.65533493528900419546138704462, 1.58724456551048532456897050947, 2.45452637871016402228825655496, 3.22754362893966258462003953821, 3.88860327572794518458625049254, 4.30040508629908295283598427180, 4.94437307110149956745580470362, 5.21166518215897327046332845629, 5.67207155470213255326225150387, 6.07128255175903051808969963153, 6.92329970739893381897680957631, 7.53997736896322143284529825653, 7.972151940820601836697648015290, 8.130186670439284459309177922112, 8.657345504637790262473996328474
