| L(s) = 1 | − 2·3-s − 4-s − 2·5-s + 3·9-s + 2·12-s + 4·15-s + 16-s + 2·20-s − 2·23-s + 3·25-s − 6·27-s − 3·36-s − 6·45-s − 2·47-s − 2·48-s + 4·49-s + 2·53-s − 2·59-s − 4·60-s − 2·64-s + 2·67-s + 4·69-s + 2·71-s − 6·75-s − 2·80-s + 9·81-s + 2·89-s + ⋯ |
| L(s) = 1 | − 2·3-s − 4-s − 2·5-s + 3·9-s + 2·12-s + 4·15-s + 16-s + 2·20-s − 2·23-s + 3·25-s − 6·27-s − 3·36-s − 6·45-s − 2·47-s − 2·48-s + 4·49-s + 2·53-s − 2·59-s − 4·60-s − 2·64-s + 2·67-s + 4·69-s + 2·71-s − 6·75-s − 2·80-s + 9·81-s + 2·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07721321722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07721321722\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.286152006837159410921583694631, −9.257399679108936937285452352229, −8.896500537691307310573059846531, −8.356505921760740358469166939759, −8.126077049385331757629198513289, −7.937765826060003358994047612184, −7.67700773859091867137421939992, −7.49654642368921679881128986129, −7.26017384000536900167833545953, −6.88681237941367835627296456317, −6.50970759583906678405247129157, −6.28222701480822907240696897913, −6.05319430970867544832352090710, −5.51610305641054235868624484660, −5.32233397059735394131619071092, −5.26195979361529551607184963417, −4.73452519105335723026706593379, −4.26725185241816332100587755687, −4.25124197981855078489667822928, −3.89465344068019022949133309138, −3.63268237617458772500686740622, −3.39534326667322651269387170485, −2.41337242316046704063508091015, −1.88468753944685974228414804407, −0.901751596604895672152761298906,
0.901751596604895672152761298906, 1.88468753944685974228414804407, 2.41337242316046704063508091015, 3.39534326667322651269387170485, 3.63268237617458772500686740622, 3.89465344068019022949133309138, 4.25124197981855078489667822928, 4.26725185241816332100587755687, 4.73452519105335723026706593379, 5.26195979361529551607184963417, 5.32233397059735394131619071092, 5.51610305641054235868624484660, 6.05319430970867544832352090710, 6.28222701480822907240696897913, 6.50970759583906678405247129157, 6.88681237941367835627296456317, 7.26017384000536900167833545953, 7.49654642368921679881128986129, 7.67700773859091867137421939992, 7.937765826060003358994047612184, 8.126077049385331757629198513289, 8.356505921760740358469166939759, 8.896500537691307310573059846531, 9.257399679108936937285452352229, 9.286152006837159410921583694631
