| L(s) = 1 | − 28·5-s − 180·13-s − 64·17-s + 392·25-s − 268·29-s − 36·37-s + 1.21e3·49-s + 1.62e3·53-s − 2.35e3·61-s + 5.04e3·65-s + 1.26e3·81-s + 1.79e3·85-s + 4.86e3·97-s + 604·101-s + 948·109-s − 7.03e3·113-s − 6.24e3·125-s + 127-s + 131-s + 137-s + 139-s + 7.50e3·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.50·5-s − 3.84·13-s − 0.913·17-s + 3.13·25-s − 1.71·29-s − 0.159·37-s + 3.53·49-s + 4.21·53-s − 4.94·61-s + 9.61·65-s + 1.73·81-s + 2.28·85-s + 5.09·97-s + 0.595·101-s + 0.833·109-s − 5.85·113-s − 4.46·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 4.29·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9369907006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9369907006\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 1262 T^{4} + p^{12} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 606 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 3209038 T^{4} + p^{12} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 90 T + 4050 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 16 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 - 89434798 T^{4} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 24254 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 134 T + 8978 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 32898 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 18 T + 162 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 p T + p^{3} T^{2} )^{2}( 1 + 10 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 3004591822 T^{4} + p^{12} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 38366 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 814 T + 331298 T^{2} - 814 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 65646280078 T^{4} + p^{12} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 1178 T + 693842 T^{2} + 1178 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 82422579218 T^{4} + p^{12} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 241502 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 166510 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 801758 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 651594598738 T^{4} + p^{12} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 1149838 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1216 T + p^{3} T^{2} )^{4} \) |
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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
−7.52446327955591466527962752594, −7.27785193714597416890670234329, −7.02336110499907030934847945585, −6.87881027281081264597880881750, −6.78537895226381582252294155515, −6.29120456380256678174106337905, −5.72658803153033367460250323147, −5.66061489569108722202138577985, −5.54461639505563337640437216631, −4.90308200853927131452359115122, −4.83287408880591062696330494354, −4.74422320261319637981188967349, −4.30019901143732261886021228806, −4.09054860516159497268936359523, −3.93032259580527610682761549003, −3.61810407828075536878244882490, −3.24963419522186573071383279743, −2.84129864921127716709083890819, −2.55685516574066886675947304813, −2.31664770560685355047153664121, −2.07405449802428428582102697677, −1.55237870453555415733387352663, −0.69683083385847460203116310939, −0.40621589848088017111930718689, −0.35549380936840880650823049417,
0.35549380936840880650823049417, 0.40621589848088017111930718689, 0.69683083385847460203116310939, 1.55237870453555415733387352663, 2.07405449802428428582102697677, 2.31664770560685355047153664121, 2.55685516574066886675947304813, 2.84129864921127716709083890819, 3.24963419522186573071383279743, 3.61810407828075536878244882490, 3.93032259580527610682761549003, 4.09054860516159497268936359523, 4.30019901143732261886021228806, 4.74422320261319637981188967349, 4.83287408880591062696330494354, 4.90308200853927131452359115122, 5.54461639505563337640437216631, 5.66061489569108722202138577985, 5.72658803153033367460250323147, 6.29120456380256678174106337905, 6.78537895226381582252294155515, 6.87881027281081264597880881750, 7.02336110499907030934847945585, 7.27785193714597416890670234329, 7.52446327955591466527962752594
