| L(s) = 1 | + 56·4-s − 162·9-s − 352·11-s + 816·16-s + 3.64e3·19-s + 5.89e3·29-s + 2.95e4·31-s − 9.07e3·36-s − 1.19e4·41-s − 1.97e4·44-s − 4.80e3·49-s − 5.25e4·59-s − 3.35e4·61-s − 2.68e4·64-s + 4.45e4·71-s + 2.04e5·76-s + 1.28e5·79-s + 1.96e4·81-s − 6.59e4·89-s + 5.70e4·99-s − 2.68e5·101-s − 2.72e5·109-s + 3.30e5·116-s + 7.32e4·121-s + 1.65e6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 7/4·4-s − 2/3·9-s − 0.877·11-s + 0.796·16-s + 2.31·19-s + 1.30·29-s + 5.51·31-s − 7/6·36-s − 1.11·41-s − 1.53·44-s − 2/7·49-s − 1.96·59-s − 1.15·61-s − 0.820·64-s + 1.04·71-s + 4.05·76-s + 2.32·79-s + 1/3·81-s − 0.882·89-s + 0.584·99-s − 2.61·101-s − 2.19·109-s + 2.27·116-s + 0.454·121-s + 9.65·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.006734391540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.006734391540\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 7 p^{3} T^{2} + 145 p^{4} T^{4} - 7 p^{13} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 16 p T + 9846 T^{2} + 16 p^{6} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 1234924 T^{2} + 654386608630 T^{4} - 1234924 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2289980 T^{2} + 5040932674246 T^{4} - 2289980 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 96 p T + 5676294 T^{2} - 96 p^{6} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 8607844 T^{2} + 63789053446 p^{2} T^{4} + 8607844 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2948 T + 32688446 T^{2} - 2948 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 14760 T + 109231790 T^{2} - 14760 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 75775604 T^{2} + 11048357286662550 T^{4} + 75775604 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 5980 T + 145983702 T^{2} + 5980 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 177795724 T^{2} + 24529024412232790 T^{4} - 177795724 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 662556924 T^{2} + 211361968232571974 T^{4} - 662556924 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 215179916 T^{2} + 357845409745732054 T^{4} - 215179916 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 26296 T + 547660454 T^{2} + 26296 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 16788 T + 1437884126 T^{2} + 16788 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 5127487084 T^{2} + 10199840530758448054 T^{4} - 5127487084 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 22264 T + 3731796926 T^{2} - 22264 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 7155352348 T^{2} + 21096538188632800486 T^{4} - 7155352348 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 64368 T + 3610665822 T^{2} - 64368 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6213307116 T^{2} + 21507760528371291254 T^{4} - 6213307116 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 32964 T + 2468206070 T^{2} + 32964 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 3942071676 T^{2} + \)\(12\!\cdots\!90\)\( T^{4} - 3942071676 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−7.03177008520787614613344633596, −6.54815848082598246842854893016, −6.47929702609690573407060893561, −6.47868481476437507506698560079, −6.26740608677147984860517219842, −5.82723301982723698801878743916, −5.57476553532503151966058743949, −5.31144716417079039927805028810, −5.06807441190289803530800181853, −4.67651885670096086281073961176, −4.62110647309647286996355648438, −4.46537279119631400554093507374, −3.83346870542661567503697670935, −3.59086682559847960648212552461, −2.96882042407220862966237129210, −2.96340709098350186017995418160, −2.88064725474949691401497018124, −2.70501590748801570998245332164, −2.34726443306323528751027130663, −1.78908196020802870564415974366, −1.73717063543218589566876434590, −0.990655869169782057467342104648, −0.971850696135024372511876033201, −0.864263861391584729187645650673, −0.00661610625317640429206983900,
0.00661610625317640429206983900, 0.864263861391584729187645650673, 0.971850696135024372511876033201, 0.990655869169782057467342104648, 1.73717063543218589566876434590, 1.78908196020802870564415974366, 2.34726443306323528751027130663, 2.70501590748801570998245332164, 2.88064725474949691401497018124, 2.96340709098350186017995418160, 2.96882042407220862966237129210, 3.59086682559847960648212552461, 3.83346870542661567503697670935, 4.46537279119631400554093507374, 4.62110647309647286996355648438, 4.67651885670096086281073961176, 5.06807441190289803530800181853, 5.31144716417079039927805028810, 5.57476553532503151966058743949, 5.82723301982723698801878743916, 6.26740608677147984860517219842, 6.47868481476437507506698560079, 6.47929702609690573407060893561, 6.54815848082598246842854893016, 7.03177008520787614613344633596
