| L(s) = 1 | + 7·4-s + 6·5-s − 18·9-s − 84·11-s + 16-s − 112·19-s + 42·20-s + 146·25-s + 636·29-s + 104·31-s − 126·36-s − 816·41-s − 588·44-s − 108·45-s + 616·49-s − 504·55-s + 372·59-s + 680·61-s + 119·64-s − 72·71-s − 784·76-s − 760·79-s + 6·80-s + 243·81-s − 2.23e3·89-s − 672·95-s + 1.51e3·99-s + ⋯ |
| L(s) = 1 | + 7/8·4-s + 0.536·5-s − 2/3·9-s − 2.30·11-s + 1/64·16-s − 1.35·19-s + 0.469·20-s + 1.16·25-s + 4.07·29-s + 0.602·31-s − 0.583·36-s − 3.10·41-s − 2.01·44-s − 0.357·45-s + 1.79·49-s − 1.23·55-s + 0.820·59-s + 1.42·61-s + 0.232·64-s − 0.120·71-s − 1.18·76-s − 1.08·79-s + 0.00838·80-s + 1/3·81-s − 2.65·89-s − 0.725·95-s + 1.53·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.045618377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045618377\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 6 T - 22 p T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 7 T^{2} + 3 p^{4} T^{4} - 7 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 88 p T^{2} + 316878 T^{4} - 88 p^{7} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 42 T + 2734 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5008 T^{2} + 13678638 T^{4} - 5008 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12400 T^{2} + 76051038 T^{4} - 12400 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 56 T + 8598 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 46204 T^{2} + 828262758 T^{4} - 46204 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 318 T + 55978 T^{2} - 318 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 52 T + 58782 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 96016 T^{2} + 4637534478 T^{4} - 96016 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 408 T + 177982 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 121900 T^{2} + 14997346998 T^{4} - 121900 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 225580 T^{2} + 31469120358 T^{4} - 225580 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 376864 T^{2} + 68697431598 T^{4} - 376864 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 186 T + 419038 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 340 T + 388398 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 861340 T^{2} + 346621507638 T^{4} - 861340 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 36 T + 384046 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 429844 T^{2} + 136740794118 T^{4} - 429844 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 380 T + 99678 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 916108 T^{2} + 434315280918 T^{4} - 916108 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 1116 T + 1508758 T^{2} + 1116 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2174980 T^{2} + 2500346420358 T^{4} - 2174980 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−14.06037495104659114004268631134, −13.97056013694459173859190634953, −13.88675440996491493311631212870, −13.12900663952029415428842181467, −12.80695705481632118945539740878, −12.72268697022107761520899556525, −11.93352692971075732567821740168, −11.79519540073729265522515856540, −11.40902190700606852083490875718, −10.58724244716950306665003513674, −10.51178912745310602829285786506, −10.26743998305704963731623170871, −10.04009475161822579189447153630, −9.082106191258618449472323379130, −8.476519061928771228668739641514, −8.271522335076912586533471794734, −8.125903885799567260777787983999, −6.91590467810850540071134234992, −6.86705334043097671359123212326, −6.37652257153146061052060194443, −5.48140434504638307387215287887, −5.17545287819315300426887780869, −4.43486437003440839041916273830, −2.79345628644283790574684224363, −2.60559007101968636034862575648,
2.60559007101968636034862575648, 2.79345628644283790574684224363, 4.43486437003440839041916273830, 5.17545287819315300426887780869, 5.48140434504638307387215287887, 6.37652257153146061052060194443, 6.86705334043097671359123212326, 6.91590467810850540071134234992, 8.125903885799567260777787983999, 8.271522335076912586533471794734, 8.476519061928771228668739641514, 9.082106191258618449472323379130, 10.04009475161822579189447153630, 10.26743998305704963731623170871, 10.51178912745310602829285786506, 10.58724244716950306665003513674, 11.40902190700606852083490875718, 11.79519540073729265522515856540, 11.93352692971075732567821740168, 12.72268697022107761520899556525, 12.80695705481632118945539740878, 13.12900663952029415428842181467, 13.88675440996491493311631212870, 13.97056013694459173859190634953, 14.06037495104659114004268631134
