| L(s) = 1 | − 32·7-s + 72·17-s + 512·23-s + 52·25-s − 160·31-s − 872·41-s + 448·47-s − 316·49-s + 4.09e3·71-s − 1.32e3·73-s + 992·79-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 + 157-s − 1.63e4·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.72·7-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 + 6.84·71-s − 2.11·73-s + 1.41·79-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 + 0.000508·157-s − 8.02·161-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.545999194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.545999194\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 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
−6.78011085632551462362499837303, −6.48557228123680577859011832239, −6.28414900835992693853561856900, −6.03312569440941834672793385632, −5.47603610827312328089621019691, −5.38731586690941028226723865844, −5.36574014281493879436250987525, −5.20283837247161520833944139718, −4.72487291536099812897088799311, −4.60988110548407017380824839463, −4.57062672916208044742490571136, −3.67641736853393534433766667374, −3.62203917414064137035769306755, −3.55003806008506280450855177536, −3.49659026108066190713414331150, −2.93960701748030304612892091760, −2.93800337824866416064038350755, −2.52323450730434796435351532634, −2.39615746850178358362119038403, −1.79359288792865149497084368777, −1.45949337423177490333232474641, −1.23075857334557121284452264106, −0.867526806968042982994519562585, −0.53958288353498521786914825917, −0.27665851115955156014433845468,
0.27665851115955156014433845468, 0.53958288353498521786914825917, 0.867526806968042982994519562585, 1.23075857334557121284452264106, 1.45949337423177490333232474641, 1.79359288792865149497084368777, 2.39615746850178358362119038403, 2.52323450730434796435351532634, 2.93800337824866416064038350755, 2.93960701748030304612892091760, 3.49659026108066190713414331150, 3.55003806008506280450855177536, 3.62203917414064137035769306755, 3.67641736853393534433766667374, 4.57062672916208044742490571136, 4.60988110548407017380824839463, 4.72487291536099812897088799311, 5.20283837247161520833944139718, 5.36574014281493879436250987525, 5.38731586690941028226723865844, 5.47603610827312328089621019691, 6.03312569440941834672793385632, 6.28414900835992693853561856900, 6.48557228123680577859011832239, 6.78011085632551462362499837303
