| L(s) = 1 | − 120·3-s − 1.35e4·5-s − 3.36e4·7-s − 2.22e5·9-s + 7.50e5·11-s − 9.54e3·13-s + 1.62e6·15-s + 4.16e6·17-s + 1.79e7·19-s + 4.03e6·21-s + 6.61e7·23-s + 3.93e7·25-s + 3.38e7·27-s + 6.15e7·29-s + 1.52e7·31-s − 9.00e7·33-s + 4.53e8·35-s − 5.27e8·37-s + 1.14e6·39-s − 1.78e8·41-s − 1.82e9·43-s + 2.99e9·45-s − 5.68e8·47-s + 8.47e8·49-s − 4.99e8·51-s − 4.18e9·53-s − 1.01e10·55-s + ⋯ |
| L(s) = 1 | − 0.285·3-s − 1.93·5-s − 0.755·7-s − 1.25·9-s + 1.40·11-s − 0.00713·13-s + 0.550·15-s + 0.710·17-s + 1.66·19-s + 0.215·21-s + 2.14·23-s + 0.806·25-s + 0.453·27-s + 0.556·29-s + 0.0958·31-s − 0.400·33-s + 1.46·35-s − 1.24·37-s + 0.00203·39-s − 0.240·41-s − 1.89·43-s + 2.42·45-s − 0.361·47-s + 3/7·49-s − 0.202·51-s − 1.37·53-s − 2.71·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 40 p T + 26290 p^{2} T^{2} + 40 p^{12} T^{3} + p^{22} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 108 p^{3} T + 5715274 p^{2} T^{2} + 108 p^{14} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 68256 p T + 653251941286 T^{2} - 68256 p^{12} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 9548 T + 580074739914 T^{2} + 9548 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4160052 T + 72837924685942 T^{2} - 4160052 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 17998712 T + 293399338219938 T^{2} - 17998712 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 66161016 T + 2994683043115918 T^{2} - 66161016 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2121228 p T + 12823193731307518 T^{2} - 2121228 p^{12} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 15281552 T + 44544265736191854 T^{2} - 15281552 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 527218340 T + 315935809764623790 T^{2} + 527218340 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 178276140 T + 1103495454485680198 T^{2} + 178276140 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1826745232 T + 2408495597390567334 T^{2} + 1826745232 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 568240704 T + 125222806434661774 T^{2} + 568240704 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4185816372 T + 8708326678160294206 T^{2} + 4185816372 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3111345000 T + 25961868574953911218 T^{2} + 3111345000 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 15042595060 T + \)\(13\!\cdots\!78\)\( T^{2} - 15042595060 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9856523968 T + \)\(12\!\cdots\!18\)\( T^{2} + 9856523968 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 24312011328 T + \)\(59\!\cdots\!02\)\( T^{2} - 24312011328 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 30890001932 T + \)\(73\!\cdots\!46\)\( T^{2} + 30890001932 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1992804256 T + \)\(85\!\cdots\!38\)\( T^{2} + 1992804256 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5277014568 T + \)\(22\!\cdots\!46\)\( T^{2} + 5277014568 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 101541312828 T + \)\(76\!\cdots\!78\)\( T^{2} + 101541312828 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 192228621116 T + \)\(23\!\cdots\!14\)\( T^{2} + 192228621116 p^{11} T^{3} + p^{22} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.22833440562733722579996382324, −11.15572182075186864905742978015, −10.01662929879969311301359759688, −9.693508411129831883316677051848, −8.829178712222627148320170411857, −8.670480833119626877407167672143, −7.898492270921304261766085989946, −7.46242258334427567041284259430, −6.68464924700331389923956088851, −6.58354871749060924721249325558, −5.30918675299710124457924719906, −5.29935098892522133448319465597, −4.21785727687163019899390854102, −3.65085684519840372367600641907, −3.15351710763104598242358011335, −2.93746249500476513881091879924, −1.40061643488169871752168687403, −0.982257066487636544453538047918, 0, 0,
0.982257066487636544453538047918, 1.40061643488169871752168687403, 2.93746249500476513881091879924, 3.15351710763104598242358011335, 3.65085684519840372367600641907, 4.21785727687163019899390854102, 5.29935098892522133448319465597, 5.30918675299710124457924719906, 6.58354871749060924721249325558, 6.68464924700331389923956088851, 7.46242258334427567041284259430, 7.898492270921304261766085989946, 8.670480833119626877407167672143, 8.829178712222627148320170411857, 9.693508411129831883316677051848, 10.01662929879969311301359759688, 11.15572182075186864905742978015, 11.22833440562733722579996382324
