| L(s) = 1 | + 3-s + 4-s + 2·7-s + 9-s + 12-s + 2·13-s + 16-s − 14·19-s + 2·21-s − 25-s + 27-s + 2·28-s − 8·31-s + 36-s − 14·37-s + 2·39-s − 2·43-s + 48-s + 3·49-s + 2·52-s − 14·57-s − 2·61-s + 2·63-s + 64-s + 28·67-s − 14·73-s − 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1/4·16-s − 3.21·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s + 0.377·28-s − 1.43·31-s + 1/6·36-s − 2.30·37-s + 0.320·39-s − 0.304·43-s + 0.144·48-s + 3/7·49-s + 0.277·52-s − 1.85·57-s − 0.256·61-s + 0.251·63-s + 1/8·64-s + 3.42·67-s − 1.63·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.231691217522472057103385997568, −7.42970330921915709586726435368, −7.16011708962485875279239287487, −6.54796581741576703271548356667, −6.45501714883769400197534752234, −5.59399109559353101448430855804, −5.38516927860402892091834425845, −4.65012025043354629223454819364, −3.98883124750352014007177515851, −3.95823828965663710949182273306, −3.13495261897234347757421228131, −2.41262911451317421726904251873, −1.89338424078267539825922227519, −1.54544973372056719849266422970, 0,
1.54544973372056719849266422970, 1.89338424078267539825922227519, 2.41262911451317421726904251873, 3.13495261897234347757421228131, 3.95823828965663710949182273306, 3.98883124750352014007177515851, 4.65012025043354629223454819364, 5.38516927860402892091834425845, 5.59399109559353101448430855804, 6.45501714883769400197534752234, 6.54796581741576703271548356667, 7.16011708962485875279239287487, 7.42970330921915709586726435368, 8.231691217522472057103385997568
