| L(s) = 1 | + 2·3-s − 3·4-s + 5-s + 3·9-s − 6·12-s + 4·13-s + 2·15-s + 5·16-s − 4·17-s − 3·20-s + 25-s + 4·27-s − 9·36-s + 8·39-s + 10·41-s + 3·45-s − 16·47-s + 10·48-s − 14·49-s − 8·51-s − 12·52-s + 20·53-s − 8·59-s − 6·60-s − 4·61-s − 3·64-s + 4·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s + 0.447·5-s + 9-s − 1.73·12-s + 1.10·13-s + 0.516·15-s + 5/4·16-s − 0.970·17-s − 0.670·20-s + 1/5·25-s + 0.769·27-s − 3/2·36-s + 1.28·39-s + 1.56·41-s + 0.447·45-s − 2.33·47-s + 1.44·48-s − 2·49-s − 1.12·51-s − 1.66·52-s + 2.74·53-s − 1.04·59-s − 0.774·60-s − 0.512·61-s − 3/8·64-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1891125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1891125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| 41 | $C_2$ | \( 1 - 10 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−7.68003844385765985083943135134, −7.37759566583217556708649005666, −6.61508786477142818172091419880, −6.35659216039682194641838348042, −5.86497653842965752491328535183, −5.31090056485434020639114264233, −4.70227288054663401602265605108, −4.52160444884874669934496061646, −3.96065425433574684505069552401, −3.58121207448309608095492288703, −3.01549784140686353642765162463, −2.50062601974282194950611376290, −1.68625435843919962684270563812, −1.18674782867264970790488068509, 0,
1.18674782867264970790488068509, 1.68625435843919962684270563812, 2.50062601974282194950611376290, 3.01549784140686353642765162463, 3.58121207448309608095492288703, 3.96065425433574684505069552401, 4.52160444884874669934496061646, 4.70227288054663401602265605108, 5.31090056485434020639114264233, 5.86497653842965752491328535183, 6.35659216039682194641838348042, 6.61508786477142818172091419880, 7.37759566583217556708649005666, 7.68003844385765985083943135134
