| L(s) = 1 | + 96·9-s − 1.09e4·13-s − 5.88e3·17-s + 6.25e3·25-s − 4.50e4·29-s − 80·37-s + 2.01e5·41-s + 1.72e5·49-s − 2.40e5·53-s + 4.60e5·61-s + 5.47e5·73-s − 5.11e5·81-s + 6.40e5·89-s − 1.10e6·97-s − 3.95e6·101-s − 5.00e6·109-s + 2.25e5·113-s − 1.05e6·117-s + 2.48e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 5.64e5·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.131·9-s − 4.98·13-s − 1.19·17-s + 2/5·25-s − 1.84·29-s − 0.00157·37-s + 2.91·41-s + 1.46·49-s − 1.61·53-s + 2.02·61-s + 1.40·73-s − 0.962·81-s + 0.908·89-s − 1.21·97-s − 3.83·101-s − 3.86·109-s + 0.156·113-s − 0.656·117-s + 1.40·121-s − 0.157·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.082254986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082254986\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 32 p T^{2} + 57854 p^{2} T^{4} - 32 p^{13} T^{6} + p^{24} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 24688 p T^{2} + 34116432366 T^{4} - 24688 p^{13} T^{6} + p^{24} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 2484244 T^{2} + 7736518816326 T^{4} - 2484244 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 5480 T + 15536718 T^{2} + 5480 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 2940 T + 27108038 T^{2} + 2940 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 6136396 p T^{2} + 6765640025292966 T^{4} - 6136396 p^{13} T^{6} + p^{24} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 351982576 T^{2} + 61525905511495086 T^{4} - 351982576 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 22524 T + 777757286 T^{2} + 22524 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 2433036724 T^{2} + 2773308609255262566 T^{4} - 2433036724 p^{12} T^{6} + p^{24} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 40 T + 2970332718 T^{2} + 40 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 100536 T + 12022755806 T^{2} - 100536 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 2475674624 T^{2} + 78196470855608993646 T^{4} + 2475674624 p^{12} T^{6} + p^{24} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 17768131696 T^{2} + \)\(31\!\cdots\!86\)\( T^{4} - 17768131696 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 120120 T + 47247961358 T^{2} + 120120 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 534190804 p T^{2} + \)\(23\!\cdots\!86\)\( T^{4} + 534190804 p^{13} T^{6} + p^{24} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 230144 T + 116150729406 T^{2} - 230144 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 341657954176 T^{2} + \)\(45\!\cdots\!66\)\( T^{4} - 341657954176 p^{12} T^{6} + p^{24} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 430053373684 T^{2} + \)\(78\!\cdots\!46\)\( T^{4} - 430053373684 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 273620 T - 4964723322 T^{2} - 273620 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 317331094084 T^{2} + \)\(72\!\cdots\!46\)\( T^{4} - 317331094084 p^{12} T^{6} + p^{24} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 429341610496 T^{2} + \)\(25\!\cdots\!26\)\( T^{4} - 429341610496 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 320244 T + 992898204806 T^{2} - 320244 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 554260 T + 457195828758 T^{2} + 554260 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−7.28346344616419572166981242060, −7.25656694159754101930205078086, −6.93666394670167377303883924568, −6.69001939127951151736185757785, −6.55558926358739484597922554856, −6.04224933308736938388128784732, −5.76622558014858029257693421515, −5.37066769214149239176526786607, −5.15388650053675625124503213198, −5.12540611842754598085529386871, −4.80613458763616417286534667067, −4.47924236910403212859125295393, −4.06959558841514624332314246055, −3.95250137139752732431381701201, −3.83023990591155153603344991951, −2.93694587308029088024375416200, −2.77631369010686486822169380620, −2.44490200537603718193957140287, −2.41718030128998740432000842346, −2.27035179590360292782681668207, −1.53696324000155765218097524402, −1.46342699873752398048650630399, −0.71207627044655364389485280536, −0.31984393949939037387743323100, −0.24939562230371212107961273219,
0.24939562230371212107961273219, 0.31984393949939037387743323100, 0.71207627044655364389485280536, 1.46342699873752398048650630399, 1.53696324000155765218097524402, 2.27035179590360292782681668207, 2.41718030128998740432000842346, 2.44490200537603718193957140287, 2.77631369010686486822169380620, 2.93694587308029088024375416200, 3.83023990591155153603344991951, 3.95250137139752732431381701201, 4.06959558841514624332314246055, 4.47924236910403212859125295393, 4.80613458763616417286534667067, 5.12540611842754598085529386871, 5.15388650053675625124503213198, 5.37066769214149239176526786607, 5.76622558014858029257693421515, 6.04224933308736938388128784732, 6.55558926358739484597922554856, 6.69001939127951151736185757785, 6.93666394670167377303883924568, 7.25656694159754101930205078086, 7.28346344616419572166981242060
