| L(s) = 1 | − 6·7-s − 3·9-s + 6·11-s − 6·17-s − 6·19-s − 12·25-s + 6·29-s − 24·31-s + 12·47-s + 9·49-s − 24·53-s − 6·59-s − 18·61-s + 18·63-s − 30·67-s − 18·71-s − 36·77-s − 6·81-s − 36·83-s − 18·99-s − 18·101-s + 30·113-s + 36·119-s − 15·121-s + 127-s + 131-s + 36·133-s + ⋯ |
| L(s) = 1 | − 2.26·7-s − 9-s + 1.80·11-s − 1.45·17-s − 1.37·19-s − 2.39·25-s + 1.11·29-s − 4.31·31-s + 1.75·47-s + 9/7·49-s − 3.29·53-s − 0.781·59-s − 2.30·61-s + 2.26·63-s − 3.66·67-s − 2.13·71-s − 4.10·77-s − 2/3·81-s − 3.95·83-s − 1.80·99-s − 1.79·101-s + 2.82·113-s + 3.30·119-s − 1.36·121-s + 0.0887·127-s + 0.0873·131-s + 3.12·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} + 5 p T^{4} + 14 p T^{6} + 5 p^{3} T^{8} + p^{5} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 + 12 T^{2} + 48 T^{4} + 118 T^{6} + 48 p^{2} T^{8} + 12 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( ( 1 + 3 T + 9 T^{2} + 26 T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 - 3 T + 21 T^{2} - 50 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( ( 1 + T + p T^{2} )^{6} \) |
| 19 | \( ( 1 + 3 T + 39 T^{2} + 78 T^{3} + 39 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 + 3 T^{2} + 327 T^{4} - 9926 T^{6} + 327 p^{2} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 3 T + 66 T^{2} - 143 T^{3} + 66 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 12 T + 117 T^{2} + 720 T^{3} + 117 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 + 141 T^{2} + 9822 T^{4} + 437033 T^{6} + 9822 p^{2} T^{8} + 141 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 + 165 T^{2} + 13350 T^{4} + 671185 T^{6} + 13350 p^{2} T^{8} + 165 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 + 159 T^{2} + 13395 T^{4} + 710714 T^{6} + 13395 p^{2} T^{8} + 159 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 6 T + 105 T^{2} - 356 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 + 12 T + 186 T^{2} + 1268 T^{3} + 186 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( ( 1 + 3 T + 105 T^{2} + 34 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 9 T + 162 T^{2} + 861 T^{3} + 162 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 + 15 T + 171 T^{2} + 1222 T^{3} + 171 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 9 T + 219 T^{2} + 1258 T^{3} + 219 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 204 T^{2} + 27696 T^{4} + 2359118 T^{6} + 27696 p^{2} T^{8} + 204 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( 1 + 78 T^{2} + 11487 T^{4} + 1040996 T^{6} + 11487 p^{2} T^{8} + 78 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 18 T + 297 T^{2} + 2860 T^{3} + 297 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 99 T^{2} + 21795 T^{4} + 1563778 T^{6} + 21795 p^{2} T^{8} + 99 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 + 267 T^{2} + 38355 T^{4} + 4179602 T^{6} + 38355 p^{2} T^{8} + 267 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.48766409520434107982067950529, −4.34258407000099000504783092455, −4.32931945686186841540505783581, −4.16109199379403991669064849817, −4.13927376154044942543925369011, −3.83811994009846207999911935253, −3.80173021184388771907555792360, −3.51996184189447659278502805493, −3.39952991168740037849350365139, −3.33143201013295558085777338226, −3.21854650908424466760777774395, −3.15266693306383504762731323887, −3.11557953033200559250174127929, −2.75496763006946786643693295169, −2.55027925300090534891806779363, −2.41083640231504220468308128770, −2.31324370298039387205154172010, −2.19925141910419266438701412431, −2.10043861106555732281903691750, −1.65270321050962222121567897004, −1.63511445230337642103434733710, −1.37328935951377395597195585054, −1.36973162213030659734144985095, −1.21544427435691185965417242963, −1.01476566759720851377389782471, 0, 0, 0, 0, 0, 0,
1.01476566759720851377389782471, 1.21544427435691185965417242963, 1.36973162213030659734144985095, 1.37328935951377395597195585054, 1.63511445230337642103434733710, 1.65270321050962222121567897004, 2.10043861106555732281903691750, 2.19925141910419266438701412431, 2.31324370298039387205154172010, 2.41083640231504220468308128770, 2.55027925300090534891806779363, 2.75496763006946786643693295169, 3.11557953033200559250174127929, 3.15266693306383504762731323887, 3.21854650908424466760777774395, 3.33143201013295558085777338226, 3.39952991168740037849350365139, 3.51996184189447659278502805493, 3.80173021184388771907555792360, 3.83811994009846207999911935253, 4.13927376154044942543925369011, 4.16109199379403991669064849817, 4.32931945686186841540505783581, 4.34258407000099000504783092455, 4.48766409520434107982067950529
Plot not available for L-functions of degree greater than 10.
