| L(s) = 1 | + 2·2-s + 3·4-s + 20·8-s − 34·9-s + 100·11-s − 75·16-s − 68·18-s + 200·22-s + 352·23-s − 314·25-s + 260·29-s − 282·32-s − 102·36-s + 212·37-s + 540·43-s + 300·44-s + 704·46-s − 628·50-s + 16·53-s + 520·58-s − 409·64-s − 1.94e3·67-s + 2.24e3·71-s − 680·72-s + 424·74-s − 1.04e3·79-s − 6·81-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 3/8·4-s + 0.883·8-s − 1.25·9-s + 2.74·11-s − 1.17·16-s − 0.890·18-s + 1.93·22-s + 3.19·23-s − 2.51·25-s + 1.66·29-s − 1.55·32-s − 0.472·36-s + 0.941·37-s + 1.91·43-s + 1.02·44-s + 2.25·46-s − 1.77·50-s + 0.0414·53-s + 1.17·58-s − 0.798·64-s − 3.54·67-s + 3.75·71-s − 1.11·72-s + 0.666·74-s − 1.49·79-s − 0.00823·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5764801 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5764801 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.789836651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.789836651\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( ( 1 - T - p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 3 | $D_4\times C_2$ | \( 1 + 34 T^{2} + 1162 T^{4} + 34 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 314 T^{2} + 48034 T^{4} + 314 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 50 T + 2702 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 5554 T^{2} + 17209282 T^{4} + 5554 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 5088 T^{2} + 11466434 T^{4} + 5088 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 12690 T^{2} + 132581162 T^{4} + 12690 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 176 T + 31038 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 130 T + 49818 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 19060 T^{2} - 9966698 T^{4} + 19060 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 106 T + 103530 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 208848 T^{2} + 19595539298 T^{4} + 208848 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 270 T + 97614 T^{2} - 270 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 228052 T^{2} + 34127693334 T^{4} + 228052 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 276710 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 632162 T^{2} + 176905491658 T^{4} + 632162 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 8978 p T^{2} + 150597445698 T^{4} + 8978 p^{7} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 972 T + 743862 T^{2} + 972 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 1124 T + 1018926 T^{2} - 1124 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 1279312 T^{2} + 705108016354 T^{4} + 1279312 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 524 T + 480382 T^{2} + 524 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 2275682 T^{2} + 1948536513034 T^{4} + 2275682 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 804000 T^{2} + 717141036962 T^{4} - 804000 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 567808 T^{2} + 211724872834 T^{4} + 567808 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−11.20257819491876418713911258748, −10.95577004290309420428988011852, −10.88046747794532624205136787868, −10.26760446400522511369056728052, −9.785279242392445464368840220259, −9.326144726423956130721730438880, −9.273059859876872027562328900671, −8.914017700696482534327261248755, −8.742629377184811453903417643264, −8.266133180865503445888015742762, −7.64099400864568382924892399128, −7.46755154542505827323762836077, −6.80570825296225348670572440666, −6.78879952285735819583181769122, −6.39820875894027246898174258689, −5.77036537804287203175257888116, −5.75166282243998565089783927738, −4.80634343790062199057737752864, −4.70001391395699142531083415200, −4.09533863556181174331652822198, −3.88051164760932429373629828840, −3.07853439106705805267098237945, −2.61622532863461852539181307197, −1.75457033817401880998895302335, −0.899811362693603773500375190422,
0.899811362693603773500375190422, 1.75457033817401880998895302335, 2.61622532863461852539181307197, 3.07853439106705805267098237945, 3.88051164760932429373629828840, 4.09533863556181174331652822198, 4.70001391395699142531083415200, 4.80634343790062199057737752864, 5.75166282243998565089783927738, 5.77036537804287203175257888116, 6.39820875894027246898174258689, 6.78879952285735819583181769122, 6.80570825296225348670572440666, 7.46755154542505827323762836077, 7.64099400864568382924892399128, 8.266133180865503445888015742762, 8.742629377184811453903417643264, 8.914017700696482534327261248755, 9.273059859876872027562328900671, 9.326144726423956130721730438880, 9.785279242392445464368840220259, 10.26760446400522511369056728052, 10.88046747794532624205136787868, 10.95577004290309420428988011852, 11.20257819491876418713911258748
