| L(s) = 1 | − 8·4-s + 66·11-s + 48·16-s − 230·19-s + 126·29-s + 590·31-s − 1.28e3·41-s − 528·44-s − 433·49-s + 2.49e3·59-s + 170·61-s − 256·64-s − 2.98e3·71-s + 1.84e3·76-s − 2.52e3·79-s + 1.36e3·89-s − 2.16e3·101-s + 4.44e3·109-s − 1.00e3·116-s − 797·121-s − 4.72e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s + 1.80·11-s + 3/4·16-s − 2.77·19-s + 0.806·29-s + 3.41·31-s − 4.89·41-s − 1.80·44-s − 1.26·49-s + 5.50·59-s + 0.356·61-s − 1/2·64-s − 4.99·71-s + 2.77·76-s − 3.60·79-s + 1.62·89-s − 2.13·101-s + 3.90·109-s − 0.806·116-s − 0.598·121-s − 3.41·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.717290858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717290858\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 433 T^{2} + 281268 T^{4} + 433 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 3 p T + 2032 T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 478 T^{2} - 5071725 T^{4} + 478 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 16547 T^{2} + 114379188 T^{4} - 16547 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 115 T + 16122 T^{2} + 115 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 145 p T^{2} + 288905028 T^{4} - 145 p^{7} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 63 T + 5560 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 295 T + 58782 T^{2} - 295 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 176711 T^{2} + 12781255848 T^{4} - 176711 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 642 T + 208402 T^{2} + 642 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 231155 T^{2} + 24116964048 T^{4} - 231155 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 94667 T^{2} + 19021837128 T^{4} - 94667 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 521312 T^{2} + 111786825390 T^{4} - 521312 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 1248 T + 742390 T^{2} - 1248 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 85 T + 447648 T^{2} - 85 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 701699 T^{2} + 296133666432 T^{4} - 701699 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1494 T + 1241350 T^{2} + 1494 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 188113 T^{2} + 283452460080 T^{4} + 188113 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 16 p T + 1353021 T^{2} + 16 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 289220 T^{2} - 150276570762 T^{4} - 289220 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 684 T + 357586 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2321471 T^{2} + 2592024266208 T^{4} - 2321471 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.36119839360652819847928809857, −6.29925760175094368640067485278, −6.24015564952804046349784647741, −5.80139484127525465601505752575, −5.71483869658418629379363684367, −5.16154140766385164422358270443, −5.02172169871310044849866523393, −4.99318277837989414115082468556, −4.61380170853469689335275566769, −4.30678583398344722015085098398, −4.29538551677891298249716620497, −4.03246099444963954370593242602, −3.89709782159346525394716737351, −3.40594477052340722495809466654, −3.35854313353701223974106608805, −2.98018699256573443830349574049, −2.70728954364549824275177257075, −2.39680267672207565907966622920, −2.12163771242230435580149080675, −1.63974010500570077866732593658, −1.55529729459022659721023039565, −1.14475341548482326804806119234, −0.994609741045932223332476787107, −0.35594656077013404304096723381, −0.24208326345877002729402555958,
0.24208326345877002729402555958, 0.35594656077013404304096723381, 0.994609741045932223332476787107, 1.14475341548482326804806119234, 1.55529729459022659721023039565, 1.63974010500570077866732593658, 2.12163771242230435580149080675, 2.39680267672207565907966622920, 2.70728954364549824275177257075, 2.98018699256573443830349574049, 3.35854313353701223974106608805, 3.40594477052340722495809466654, 3.89709782159346525394716737351, 4.03246099444963954370593242602, 4.29538551677891298249716620497, 4.30678583398344722015085098398, 4.61380170853469689335275566769, 4.99318277837989414115082468556, 5.02172169871310044849866523393, 5.16154140766385164422358270443, 5.71483869658418629379363684367, 5.80139484127525465601505752575, 6.24015564952804046349784647741, 6.29925760175094368640067485278, 6.36119839360652819847928809857
