| L(s) = 1 | + 13·4-s + 36·7-s − 52·13-s + 57·16-s + 84·19-s + 80·25-s + 468·28-s − 604·31-s + 184·37-s + 880·43-s − 96·49-s − 676·52-s + 656·61-s + 117·64-s + 3.05e3·67-s − 312·73-s + 1.09e3·76-s − 720·79-s − 1.87e3·91-s − 344·97-s + 1.04e3·100-s + 3.39e3·103-s + 3.38e3·109-s + 2.05e3·112-s − 4.57e3·121-s − 7.85e3·124-s + 127-s + ⋯ |
| L(s) = 1 | + 13/8·4-s + 1.94·7-s − 1.10·13-s + 0.890·16-s + 1.01·19-s + 0.639·25-s + 3.15·28-s − 3.49·31-s + 0.817·37-s + 3.12·43-s − 0.279·49-s − 1.80·52-s + 1.37·61-s + 0.228·64-s + 5.56·67-s − 0.500·73-s + 1.64·76-s − 1.02·79-s − 2.15·91-s − 0.360·97-s + 1.03·100-s + 3.24·103-s + 2.97·109-s + 1.73·112-s − 3.43·121-s − 5.68·124-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.757908317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.757908317\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 - 13 T^{2} + 7 p^{4} T^{4} - 13 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 16 p T^{2} + 24462 T^{4} - 16 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 18 T + 534 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 4572 T^{2} + 8634710 T^{4} + 4572 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 16772 T^{2} + 118361542 T^{4} + 16772 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 42 T + 2742 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 12 T^{2} - 393658 T^{4} + 12 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 22356 T^{2} - 27485674 T^{4} + 22356 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 302 T + 2650 p T^{2} + 302 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 92 T + 43774 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 227840 T^{2} + 22474730590 T^{4} + 227840 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 440 T + 203686 T^{2} - 440 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 270652 T^{2} + 39420081222 T^{4} + 270652 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 438372 T^{2} + 89997219542 T^{4} + 438372 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 598556 T^{2} + 161876495254 T^{4} + 598556 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 328 T + 368086 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 1526 T + 1116358 T^{2} - 1526 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 500316 T^{2} + 133803927398 T^{4} + 500316 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 156 T - 293274 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 360 T + 287790 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 2272508 T^{2} + 1944939568822 T^{4} + 2272508 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 913024 T^{2} + 920290760478 T^{4} + 913024 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 172 T + 1202710 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−9.723349098011715884915672436369, −9.157802729041155800550253748225, −8.888707946839727007921249446855, −8.620282655146591122367847028695, −8.140572683382447779593292704795, −7.902742700429722967954113724811, −7.74635422064524179828962271735, −7.37998279142584114963315877880, −7.09471368874207300444231247811, −6.96138103094564345573001950821, −6.76251208983625190013642119656, −5.95466041927721683549380325541, −5.92184992788502698800587133080, −5.46475534796872290943980054497, −5.18963424150428009812672372367, −4.85995673080021990377587814525, −4.56525955689829671607952728377, −4.00599306635169217245363092163, −3.54762667186843731435185883866, −3.21868774675921166154517358044, −2.37676234701599773990078476292, −2.20702796588499446205262391721, −2.02992181189936806077436563041, −1.27146965433305099623107560549, −0.67484081611758928457459810541,
0.67484081611758928457459810541, 1.27146965433305099623107560549, 2.02992181189936806077436563041, 2.20702796588499446205262391721, 2.37676234701599773990078476292, 3.21868774675921166154517358044, 3.54762667186843731435185883866, 4.00599306635169217245363092163, 4.56525955689829671607952728377, 4.85995673080021990377587814525, 5.18963424150428009812672372367, 5.46475534796872290943980054497, 5.92184992788502698800587133080, 5.95466041927721683549380325541, 6.76251208983625190013642119656, 6.96138103094564345573001950821, 7.09471368874207300444231247811, 7.37998279142584114963315877880, 7.74635422064524179828962271735, 7.902742700429722967954113724811, 8.140572683382447779593292704795, 8.620282655146591122367847028695, 8.888707946839727007921249446855, 9.157802729041155800550253748225, 9.723349098011715884915672436369
