L(s) = 1 | − 16·4-s − 16·5-s + 124·16-s + 256·20-s − 136·23-s + 140·25-s − 204·31-s − 172·37-s − 664·47-s − 198·49-s − 2.52e3·53-s − 360·59-s − 896·64-s + 2.62e3·67-s − 512·71-s − 1.98e3·80-s + 544·89-s + 2.17e3·92-s + 20·97-s − 2.24e3·100-s − 6.90e3·103-s + 1.41e3·113-s + 2.17e3·115-s + 3.26e3·124-s − 2.80e3·125-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 2·4-s − 1.43·5-s + 1.93·16-s + 2.86·20-s − 1.23·23-s + 1.11·25-s − 1.18·31-s − 0.764·37-s − 2.06·47-s − 0.577·49-s − 6.53·53-s − 0.794·59-s − 7/4·64-s + 4.79·67-s − 0.855·71-s − 2.77·80-s + 0.647·89-s + 2.46·92-s + 0.0209·97-s − 2.23·100-s − 6.60·103-s + 1.17·113-s + 1.76·115-s + 2.36·124-s − 2.00·125-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
good | 2 | $D_4\times C_2$ | \( 1 + p^{4} T^{2} + 33 p^{2} T^{4} + p^{10} T^{6} + p^{12} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 8 T + 26 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 198 T^{2} + 1691 p^{2} T^{4} + 198 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 2644 T^{2} + 9189462 T^{4} + 2644 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 9412 T^{2} + 64277574 T^{4} + 9412 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 14310 T^{2} + 108781787 T^{4} + 14310 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 68 T + 16850 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 13508 T^{2} + 921166518 T^{4} - 13508 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 102 T + 50423 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 86 T + 25395 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 41068 T^{2} + 4308998598 T^{4} + 41068 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 284532 T^{2} + 32602404854 T^{4} + 284532 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 332 T + 223442 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 1260 T + 686014 T^{2} + 1260 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 180 T + 410218 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 431934 T^{2} + 130927549811 T^{4} + 431934 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 1314 T + 979175 T^{2} - 1314 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 256 T + 726206 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 754158 T^{2} + 442810688819 T^{4} + 754158 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1133782 T^{2} + 749178836283 T^{4} + 1133782 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 2145652 T^{2} + 1800431735574 T^{4} + 2145652 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 272 T + 1341794 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 10 T + 1176411 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.97979090553143170121881693109, −6.73817769949364593862036326085, −6.72575530612860517037791961391, −6.65142631394627343520700281158, −6.23621015485290140263276760889, −5.85509633265108492289288182171, −5.66642169604484636959783776529, −5.60474641551704351490334293999, −5.05538888296220829592984957359, −4.88518062068692837565217709924, −4.80178632594884456857314762494, −4.72804915134967849892178615435, −4.40641780847895113845445641940, −4.04850240292724364047428441992, −3.80250867870284271448645481379, −3.67706730458899000721820854023, −3.55719979478720578816988712799, −3.16425964904722854996247320129, −2.86664504811606822795833752540, −2.78841461022219086998388172729, −1.98330718328353880875219832066, −1.84389508714957598428679801367, −1.58271270368584878001513198021, −1.14189994465270268898223411170, −0.841024857757212211429444987996, 0, 0, 0, 0,
0.841024857757212211429444987996, 1.14189994465270268898223411170, 1.58271270368584878001513198021, 1.84389508714957598428679801367, 1.98330718328353880875219832066, 2.78841461022219086998388172729, 2.86664504811606822795833752540, 3.16425964904722854996247320129, 3.55719979478720578816988712799, 3.67706730458899000721820854023, 3.80250867870284271448645481379, 4.04850240292724364047428441992, 4.40641780847895113845445641940, 4.72804915134967849892178615435, 4.80178632594884456857314762494, 4.88518062068692837565217709924, 5.05538888296220829592984957359, 5.60474641551704351490334293999, 5.66642169604484636959783776529, 5.85509633265108492289288182171, 6.23621015485290140263276760889, 6.65142631394627343520700281158, 6.72575530612860517037791961391, 6.73817769949364593862036326085, 6.97979090553143170121881693109