| L(s) = 1 | − 8·3-s + 4·9-s − 16·11-s − 56·17-s + 40·19-s + 36·25-s + 200·27-s + 128·33-s + 120·41-s + 48·43-s − 14·49-s + 448·51-s − 320·57-s − 72·59-s − 96·67-s + 152·73-s − 288·75-s − 790·81-s − 248·83-s − 136·89-s − 600·97-s − 64·99-s − 352·107-s + 56·113-s − 100·121-s − 960·123-s + 127-s + ⋯ |
| L(s) = 1 | − 8/3·3-s + 4/9·9-s − 1.45·11-s − 3.29·17-s + 2.10·19-s + 1.43·25-s + 7.40·27-s + 3.87·33-s + 2.92·41-s + 1.11·43-s − 2/7·49-s + 8.78·51-s − 5.61·57-s − 1.22·59-s − 1.43·67-s + 2.08·73-s − 3.83·75-s − 9.75·81-s − 2.98·83-s − 1.52·89-s − 6.18·97-s − 0.646·99-s − 3.28·107-s + 0.495·113-s − 0.826·121-s − 7.80·123-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2349406042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2349406042\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 5 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1126 T^{4} - 36 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 146 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 59814 T^{4} - 164 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 28 T + 662 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 20 T + 374 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1604 T^{2} + 1138374 T^{4} - 1604 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1148 T^{2} + 1340838 T^{4} - 1148 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 1906 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2940 T^{2} + 5391334 T^{4} - 2940 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 60 T + 4150 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 24 T + 1042 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 5348 T^{2} + 14587206 T^{4} - 5348 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 40 T + 10 p T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )( 1 + 40 T + 10 p T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 59 | $D_{4}$ | \( ( 1 + 36 T + 5494 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14372 T^{2} + 79326246 T^{4} - 14372 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 48 T + 6754 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 8388 T^{2} + 39052870 T^{4} - 8388 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 76 T + 4934 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 1282 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 124 T + 13590 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 68 T + 15206 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 300 T + 41206 T^{2} + 300 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.42988819351148331803952381540, −6.24150135156330618320986921558, −5.80076762655698813949561142677, −5.52758336073885517331226457377, −5.51103623253157342463379289688, −5.48641276197376313645578530726, −5.47786017711144637461090227884, −5.02204698586488490505416687774, −4.78484205559981225436075137526, −4.51364040493759642281148607530, −4.37910050143639970520278864036, −4.22339352199955153377541884669, −3.96940406289386693608477611022, −3.44766762437138045057661993602, −2.90163188304610637926107566354, −2.89058407351091158621048909971, −2.65426650314275188619336222612, −2.63704523955576335294865536738, −2.59821704406584456937715114089, −1.74359921270576799566684800566, −1.47058175943931424040885457711, −1.07345681975946491705963969418, −0.59098298775835572997573372868, −0.46647094489569426834829471112, −0.17393457908813333250237073901,
0.17393457908813333250237073901, 0.46647094489569426834829471112, 0.59098298775835572997573372868, 1.07345681975946491705963969418, 1.47058175943931424040885457711, 1.74359921270576799566684800566, 2.59821704406584456937715114089, 2.63704523955576335294865536738, 2.65426650314275188619336222612, 2.89058407351091158621048909971, 2.90163188304610637926107566354, 3.44766762437138045057661993602, 3.96940406289386693608477611022, 4.22339352199955153377541884669, 4.37910050143639970520278864036, 4.51364040493759642281148607530, 4.78484205559981225436075137526, 5.02204698586488490505416687774, 5.47786017711144637461090227884, 5.48641276197376313645578530726, 5.51103623253157342463379289688, 5.52758336073885517331226457377, 5.80076762655698813949561142677, 6.24150135156330618320986921558, 6.42988819351148331803952381540
