L(s) = 1 | − 2·3-s − 5-s + 3·9-s + 2·15-s + 25-s − 4·27-s − 4·31-s − 4·37-s − 4·43-s − 3·45-s + 6·49-s + 4·53-s − 8·67-s + 12·71-s − 2·75-s + 4·79-s + 5·81-s − 12·83-s + 8·93-s − 28·107-s + 8·111-s − 2·121-s − 125-s + 127-s + 8·129-s + 131-s + 4·135-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.447·5-s + 9-s + 0.516·15-s + 1/5·25-s − 0.769·27-s − 0.718·31-s − 0.657·37-s − 0.609·43-s − 0.447·45-s + 6/7·49-s + 0.549·53-s − 0.977·67-s + 1.42·71-s − 0.230·75-s + 0.450·79-s + 5/9·81-s − 1.31·83-s + 0.829·93-s − 2.70·107-s + 0.759·111-s − 0.181·121-s − 0.0894·125-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.344·135-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
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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
−8.467098054064982480192351096104, −8.232731172496917014237011424936, −7.54377415676442119100095116130, −7.09995978847628383742192600281, −6.81080123504408923347856299000, −6.19537445648966642748097383156, −5.69815290434057809862999349532, −5.24974680545753486374005468927, −4.80208967704798766844481283023, −4.11740723102919499188580276115, −3.73391769766970712671254778299, −2.94189892048369421044102755305, −2.04408330697429802940421563950, −1.14592922526088296649228663892, 0,
1.14592922526088296649228663892, 2.04408330697429802940421563950, 2.94189892048369421044102755305, 3.73391769766970712671254778299, 4.11740723102919499188580276115, 4.80208967704798766844481283023, 5.24974680545753486374005468927, 5.69815290434057809862999349532, 6.19537445648966642748097383156, 6.81080123504408923347856299000, 7.09995978847628383742192600281, 7.54377415676442119100095116130, 8.232731172496917014237011424936, 8.467098054064982480192351096104