L(s) = 1 | − 2·3-s + 3·9-s + 8·11-s − 4·17-s − 8·19-s − 6·25-s − 4·27-s − 16·33-s − 4·41-s − 16·43-s + 49-s + 8·51-s + 16·57-s + 24·59-s + 16·67-s + 4·73-s + 12·75-s + 5·81-s − 24·83-s + 12·89-s + 4·97-s + 24·99-s + 24·107-s − 28·113-s + 26·121-s + 8·123-s + 127-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s + 2.41·11-s − 0.970·17-s − 1.83·19-s − 6/5·25-s − 0.769·27-s − 2.78·33-s − 0.624·41-s − 2.43·43-s + 1/7·49-s + 1.12·51-s + 2.11·57-s + 3.12·59-s + 1.95·67-s + 0.468·73-s + 1.38·75-s + 5/9·81-s − 2.63·83-s + 1.27·89-s + 0.406·97-s + 2.41·99-s + 2.32·107-s − 2.63·113-s + 2.36·121-s + 0.721·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.190173500954779681525780731187, −7.20214522889553119059062687435, −6.78044077445841975719051191645, −6.66339990127808429560142628788, −6.35837082667327210170937090345, −5.74243037997109166467332435643, −5.36857771747617040863256466681, −4.52969287471330622027217048773, −4.43219841019389924830737127106, −3.69109001567993476839628496737, −3.61417461907435968748262507791, −2.19510980306732737447197047942, −1.90612310330978239121510440566, −1.06346104514984424021583213227, 0,
1.06346104514984424021583213227, 1.90612310330978239121510440566, 2.19510980306732737447197047942, 3.61417461907435968748262507791, 3.69109001567993476839628496737, 4.43219841019389924830737127106, 4.52969287471330622027217048773, 5.36857771747617040863256466681, 5.74243037997109166467332435643, 6.35837082667327210170937090345, 6.66339990127808429560142628788, 6.78044077445841975719051191645, 7.20214522889553119059062687435, 8.190173500954779681525780731187