| L(s) = 1 | − 32·7-s − 18·9-s − 72·17-s − 512·23-s + 52·25-s − 160·31-s + 872·41-s − 448·47-s − 316·49-s + 576·63-s − 4.09e3·71-s − 1.32e3·73-s + 992·79-s + 243·81-s − 1.06e3·89-s − 2.44e3·97-s − 3.42e3·103-s + 456·113-s + 2.30e3·119-s − 1.36e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.29e3·153-s + ⋯ |
| L(s) = 1 | − 1.72·7-s − 2/3·9-s − 1.02·17-s − 4.64·23-s + 0.415·25-s − 0.926·31-s + 3.32·41-s − 1.39·47-s − 0.921·49-s + 1.15·63-s − 6.84·71-s − 2.11·73-s + 1.41·79-s + 1/3·81-s − 1.26·89-s − 2.55·97-s − 3.27·103-s + 0.379·113-s + 1.77·119-s − 1.02·121-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 + 0.684·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2744806734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2744806734\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 18614 T^{4} - 52 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 16 T + 542 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 124 p T^{2} + 3795254 T^{4} + 124 p^{7} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4532 T^{2} + 10475286 T^{4} - 4532 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 36 T + 6822 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 11532 T^{2} + 65783830 T^{4} - 11532 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 256 T + 39886 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 52308 T^{2} + 1484422166 T^{4} - 52308 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 80 T + 26030 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 186708 T^{2} + 13784867446 T^{4} - 186708 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 436 T + 102166 T^{2} - 436 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 57900 T^{2} + 11793718966 T^{4} - 57900 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 224 T + 179422 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 442740 T^{2} + 87795274358 T^{4} - 442740 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 305782 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 348986 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 140620 T^{2} + 86210675286 T^{4} - 140620 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 2048 T + 1763566 T^{2} + 2048 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 660 T + 407702 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 496 T + 323534 T^{2} - 496 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1659916 T^{2} + 1244512983830 T^{4} - 1659916 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 532 T + 1467382 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 1220 T + 586694 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.87956553889069554842887383892, −7.42016673420294087907256961587, −7.36411157089547635280624042148, −6.89109352122068319829208800904, −6.72161662339250524141154467883, −6.60010110586515617896198051441, −6.01994780541879112933061756456, −5.91221435244200454627231044279, −5.90376624421957540451842809138, −5.58186394336659845840260518608, −5.57822643133858834751726585092, −4.56423367039094917598139386040, −4.41624287453501589321172149913, −4.30330157251474642893249269433, −4.28532495447804495880265413365, −3.66239675587990975576090768560, −3.27529135064497163698833997620, −3.08871217206903087785157406804, −2.83060790398305667493168365201, −2.44114199015463559474032336621, −1.92185296600064845346543788191, −1.77844431919234518120859639023, −1.28493397482986512940006365712, −0.24560500845527334316821758714, −0.24207730055192146881997137140,
0.24207730055192146881997137140, 0.24560500845527334316821758714, 1.28493397482986512940006365712, 1.77844431919234518120859639023, 1.92185296600064845346543788191, 2.44114199015463559474032336621, 2.83060790398305667493168365201, 3.08871217206903087785157406804, 3.27529135064497163698833997620, 3.66239675587990975576090768560, 4.28532495447804495880265413365, 4.30330157251474642893249269433, 4.41624287453501589321172149913, 4.56423367039094917598139386040, 5.57822643133858834751726585092, 5.58186394336659845840260518608, 5.90376624421957540451842809138, 5.91221435244200454627231044279, 6.01994780541879112933061756456, 6.60010110586515617896198051441, 6.72161662339250524141154467883, 6.89109352122068319829208800904, 7.36411157089547635280624042148, 7.42016673420294087907256961587, 7.87956553889069554842887383892
