| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 3·8-s − 3·9-s − 4·11-s + 2·12-s − 16-s − 12·17-s + 3·18-s + 8·19-s + 4·22-s − 6·24-s − 25-s + 14·27-s − 5·32-s + 8·33-s + 12·34-s + 3·36-s − 8·38-s − 20·41-s + 8·43-s + 4·44-s + 2·48-s − 13·49-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 1.06·8-s − 9-s − 1.20·11-s + 0.577·12-s − 1/4·16-s − 2.91·17-s + 0.707·18-s + 1.83·19-s + 0.852·22-s − 1.22·24-s − 1/5·25-s + 2.69·27-s − 0.883·32-s + 1.39·33-s + 2.05·34-s + 1/2·36-s − 1.29·38-s − 3.12·41-s + 1.21·43-s + 0.603·44-s + 0.288·48-s − 1.85·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 19 T + p T^{2} )^{2} \) |
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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
−8.282431792555002607496667489786, −8.050361973857519577364561062135, −7.12147698344619796195085066396, −6.85141315820698144674268214913, −6.41485281254561031078125651012, −5.66518849949962625096457022893, −5.29917071437467895295333587611, −4.99967154849558777078789522945, −4.55722418726185892367712037207, −3.76592143771251579951082054976, −2.94012216377286837608157805866, −2.44967157596829835846815768154, −1.41011996508034289590507323013, 0, 0,
1.41011996508034289590507323013, 2.44967157596829835846815768154, 2.94012216377286837608157805866, 3.76592143771251579951082054976, 4.55722418726185892367712037207, 4.99967154849558777078789522945, 5.29917071437467895295333587611, 5.66518849949962625096457022893, 6.41485281254561031078125651012, 6.85141315820698144674268214913, 7.12147698344619796195085066396, 8.050361973857519577364561062135, 8.282431792555002607496667489786
