| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 4·13-s + 5·16-s − 4·19-s + 6·23-s − 10·25-s − 8·26-s − 10·31-s + 6·32-s − 8·37-s − 8·38-s − 6·41-s + 4·43-s + 12·46-s − 18·47-s − 20·50-s − 12·52-s − 4·61-s − 20·62-s + 7·64-s − 8·67-s − 6·71-s + 14·73-s − 16·74-s − 12·76-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 1.10·13-s + 5/4·16-s − 0.917·19-s + 1.25·23-s − 2·25-s − 1.56·26-s − 1.79·31-s + 1.06·32-s − 1.31·37-s − 1.29·38-s − 0.937·41-s + 0.609·43-s + 1.76·46-s − 2.62·47-s − 2.82·50-s − 1.66·52-s − 0.512·61-s − 2.54·62-s + 7/8·64-s − 0.977·67-s − 0.712·71-s + 1.63·73-s − 1.85·74-s − 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 157 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 108 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 133 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 165 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 292 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 115 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.40971043247299713100666432346, −7.29380536921522461791324028112, −6.80202217732845795939464729903, −6.74760429291660929004530531662, −6.17269066006716200603389953134, −5.86477322032396701234545869710, −5.45608072161696427135646757883, −5.24784604285028041449704434931, −4.74922958778590638718419471749, −4.67343780110178242937896718250, −3.99599631547843257542725389735, −3.84922054848752280456182446681, −3.22570622191344067617966086721, −3.20595472947644881187796041516, −2.36391495039277332205872234668, −2.28799448991214395916509928292, −1.53912586395349666517643384759, −1.48756799661581509278497207182, 0, 0,
1.48756799661581509278497207182, 1.53912586395349666517643384759, 2.28799448991214395916509928292, 2.36391495039277332205872234668, 3.20595472947644881187796041516, 3.22570622191344067617966086721, 3.84922054848752280456182446681, 3.99599631547843257542725389735, 4.67343780110178242937896718250, 4.74922958778590638718419471749, 5.24784604285028041449704434931, 5.45608072161696427135646757883, 5.86477322032396701234545869710, 6.17269066006716200603389953134, 6.74760429291660929004530531662, 6.80202217732845795939464729903, 7.29380536921522461791324028112, 7.40971043247299713100666432346
