| L(s) = 1 | − 8·5-s − 40·13-s + 24·17-s + 36·25-s − 40·29-s − 72·37-s + 88·41-s + 68·49-s + 24·53-s + 56·61-s + 320·65-s − 120·73-s − 192·85-s + 312·89-s − 248·97-s + 664·101-s + 24·109-s − 328·113-s + 260·121-s − 312·125-s + 127-s + 131-s + 137-s + 139-s + 320·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 8/5·5-s − 3.07·13-s + 1.41·17-s + 1.43·25-s − 1.37·29-s − 1.94·37-s + 2.14·41-s + 1.38·49-s + 0.452·53-s + 0.918·61-s + 4.92·65-s − 1.64·73-s − 2.25·85-s + 3.50·89-s − 2.55·97-s + 6.57·101-s + 0.220·109-s − 2.90·113-s + 2.14·121-s − 2.49·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 2.20·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8465691045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8465691045\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 + 4 T + 6 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2886 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 33894 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 20 T + 246 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 12 T + 422 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 196 T^{2} + 159654 T^{4} - 196 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 994 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 20 T + 1734 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1412 T^{2} + 2268678 T^{4} - 1412 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 36 T + 62 p T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 44 T + 3654 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1796 T^{2} - 35994 T^{4} - 1796 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2180 T^{2} + 8539014 T^{4} - 2180 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 5606 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 462054 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 11204 T^{2} + 62050854 T^{4} - 11204 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3140 T^{2} + 9051462 T^{4} - 3140 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 60 T + 8486 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12548 T^{2} + 107282310 T^{4} - 12548 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 21188 T^{2} + 206547366 T^{4} - 21188 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 156 T + 21158 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 124 T + 15750 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.52438374954939539066739658922, −7.39374203522856472916911074083, −7.33111972673892842354723970812, −6.80715483794219014320216838834, −6.70115426040226872500102755743, −6.48358593305154638596442012125, −5.96788742161020360550873752533, −5.61702397886775138779214208748, −5.57953712735072537285436821872, −5.35701101692441262785952276464, −4.90142185211872773313258869980, −4.79957337961246133709169412238, −4.57201151473181446758171099494, −4.21208103217791599690275912385, −4.00275228725056824643976713697, −3.53621501352900788058153980708, −3.34529455010938173926051504705, −3.33939308054023229405815574213, −2.60168397715215953606669668367, −2.39915259726407197359768972463, −2.27648725667757689656931121595, −1.72083659858777630761953245313, −1.11097786787057794774085387329, −0.61602206945630519824770054554, −0.24871919521653053427727555090,
0.24871919521653053427727555090, 0.61602206945630519824770054554, 1.11097786787057794774085387329, 1.72083659858777630761953245313, 2.27648725667757689656931121595, 2.39915259726407197359768972463, 2.60168397715215953606669668367, 3.33939308054023229405815574213, 3.34529455010938173926051504705, 3.53621501352900788058153980708, 4.00275228725056824643976713697, 4.21208103217791599690275912385, 4.57201151473181446758171099494, 4.79957337961246133709169412238, 4.90142185211872773313258869980, 5.35701101692441262785952276464, 5.57953712735072537285436821872, 5.61702397886775138779214208748, 5.96788742161020360550873752533, 6.48358593305154638596442012125, 6.70115426040226872500102755743, 6.80715483794219014320216838834, 7.33111972673892842354723970812, 7.39374203522856472916911074083, 7.52438374954939539066739658922
