| L(s) = 1 | − 2·3-s − 2·7-s + 3·9-s − 4·13-s − 4·17-s + 4·21-s − 4·27-s − 4·29-s + 8·31-s − 12·37-s + 8·39-s + 4·41-s + 8·43-s + 16·47-s + 3·49-s + 8·51-s − 4·53-s − 8·59-s + 12·61-s − 6·63-s + 8·67-s + 8·71-s − 12·73-s + 5·81-s + 8·83-s + 8·87-s − 28·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 9-s − 1.10·13-s − 0.970·17-s + 0.872·21-s − 0.769·27-s − 0.742·29-s + 1.43·31-s − 1.97·37-s + 1.28·39-s + 0.624·41-s + 1.21·43-s + 2.33·47-s + 3/7·49-s + 1.12·51-s − 0.549·53-s − 1.04·59-s + 1.53·61-s − 0.755·63-s + 0.977·67-s + 0.949·71-s − 1.40·73-s + 5/9·81-s + 0.878·83-s + 0.857·87-s − 2.96·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.124307538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.124307538\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.601391745108637722899934915740, −8.304198681608398962245151514098, −7.50973846425111185570099065834, −7.49267321886468964500898938259, −7.09496358704795333893016945820, −6.73062210066174641407530229062, −6.40613991462151765412235079546, −6.01978476215382816423381247562, −5.53575991027225031852163147678, −5.46569926755382447838056336673, −4.72690889928127726949864532961, −4.64889618335368037927417033431, −3.97118092484213871507013925884, −3.90126385717873024788387985266, −3.03854100236757681317097516293, −2.75757354697635502760111690403, −2.11460088127017549079664598427, −1.78487256949889422398927118715, −0.75955387726228538778460445762, −0.46226328566823296926696376755,
0.46226328566823296926696376755, 0.75955387726228538778460445762, 1.78487256949889422398927118715, 2.11460088127017549079664598427, 2.75757354697635502760111690403, 3.03854100236757681317097516293, 3.90126385717873024788387985266, 3.97118092484213871507013925884, 4.64889618335368037927417033431, 4.72690889928127726949864532961, 5.46569926755382447838056336673, 5.53575991027225031852163147678, 6.01978476215382816423381247562, 6.40613991462151765412235079546, 6.73062210066174641407530229062, 7.09496358704795333893016945820, 7.49267321886468964500898938259, 7.50973846425111185570099065834, 8.304198681608398962245151514098, 8.601391745108637722899934915740
