| L(s) = 1 | + 2-s − 4-s − 3·8-s − 6·9-s − 16-s + 12·17-s − 6·18-s + 8·23-s + 25-s − 16·31-s + 5·32-s + 12·34-s + 6·36-s + 4·41-s + 8·46-s − 24·47-s − 14·49-s + 50-s − 16·62-s + 7·64-s − 12·68-s + 16·71-s + 18·72-s + 28·73-s + 16·79-s + 27·81-s + 4·82-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 2·9-s − 1/4·16-s + 2.91·17-s − 1.41·18-s + 1.66·23-s + 1/5·25-s − 2.87·31-s + 0.883·32-s + 2.05·34-s + 36-s + 0.624·41-s + 1.17·46-s − 3.50·47-s − 2·49-s + 0.141·50-s − 2.03·62-s + 7/8·64-s − 1.45·68-s + 1.89·71-s + 2.12·72-s + 3.27·73-s + 1.80·79-s + 3·81-s + 0.441·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.496466589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496466589\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−9.365267708430525132762162914792, −8.448904786013577857416673383279, −8.248642768475844318802620430882, −7.84109121338928955235594960129, −7.22566483951161934358497324239, −6.34700070707920035984611739420, −6.16821463973303244938343944630, −5.33894927898579958226937092107, −5.09766059609940450216711100862, −5.05872363288651094885006908150, −3.56946334592753091691191901249, −3.29934527379638532175228783242, −3.25423179226253980082861279896, −2.03779058571190597135714039078, −0.70358892174385003118951574686,
0.70358892174385003118951574686, 2.03779058571190597135714039078, 3.25423179226253980082861279896, 3.29934527379638532175228783242, 3.56946334592753091691191901249, 5.05872363288651094885006908150, 5.09766059609940450216711100862, 5.33894927898579958226937092107, 6.16821463973303244938343944630, 6.34700070707920035984611739420, 7.22566483951161934358497324239, 7.84109121338928955235594960129, 8.248642768475844318802620430882, 8.448904786013577857416673383279, 9.365267708430525132762162914792
