| L(s) = 1 | + 2·3-s − 4-s + 9-s − 2·12-s − 3·16-s − 4·27-s − 8·31-s − 36-s + 20·37-s − 6·48-s + 2·49-s + 7·64-s + 20·67-s − 11·81-s − 16·93-s + 20·97-s − 4·103-s + 4·108-s + 40·111-s − 11·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 4·147-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 1/3·9-s − 0.577·12-s − 3/4·16-s − 0.769·27-s − 1.43·31-s − 1/6·36-s + 3.28·37-s − 0.866·48-s + 2/7·49-s + 7/8·64-s + 2.44·67-s − 1.22·81-s − 1.65·93-s + 2.03·97-s − 0.394·103-s + 0.384·108-s + 3.79·111-s − 121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + 0.329·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.220925642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.220925642\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.270739097386057173767019349661, −7.958494441113673965788217617984, −7.66928656608797906504558359964, −7.07266225861375126234499501264, −6.65025047071107930531917789015, −6.01685878491248147659667464463, −5.63793667687400664861302180846, −4.98938860487290086047655463414, −4.51373414916300345266394886519, −3.92019977328633530943646916412, −3.66010568298275388161515661623, −2.83064468220402663839407493896, −2.43664644666960566585683400247, −1.82616425751355944134602451145, −0.68895182792238013690135482511,
0.68895182792238013690135482511, 1.82616425751355944134602451145, 2.43664644666960566585683400247, 2.83064468220402663839407493896, 3.66010568298275388161515661623, 3.92019977328633530943646916412, 4.51373414916300345266394886519, 4.98938860487290086047655463414, 5.63793667687400664861302180846, 6.01685878491248147659667464463, 6.65025047071107930531917789015, 7.07266225861375126234499501264, 7.66928656608797906504558359964, 7.958494441113673965788217617984, 8.270739097386057173767019349661
