| L(s) = 1 | + 2·3-s − 7·4-s + 55·9-s − 14·12-s + 64·16-s + 90·17-s − 324·23-s + 338·25-s + 214·27-s + 288·29-s − 385·36-s + 194·43-s + 128·48-s − 461·49-s + 180·51-s − 1.65e3·53-s − 752·61-s − 1.00e3·64-s − 630·68-s − 648·69-s + 676·75-s − 3.32e3·79-s + 1.04e3·81-s + 576·87-s + 2.26e3·92-s − 2.36e3·100-s − 792·101-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 7/8·4-s + 2.03·9-s − 0.336·12-s + 16-s + 1.28·17-s − 2.93·23-s + 2.70·25-s + 1.52·27-s + 1.84·29-s − 1.78·36-s + 0.688·43-s + 0.384·48-s − 1.34·49-s + 0.494·51-s − 4.29·53-s − 1.57·61-s − 1.95·64-s − 1.12·68-s − 1.13·69-s + 1.04·75-s − 4.72·79-s + 1.43·81-s + 0.709·87-s + 2.57·92-s − 2.36·100-s − 0.780·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.493975456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.493975456\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 7 T^{2} - 15 T^{4} + 7 p^{6} T^{6} + p^{12} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 - T - 26 T^{2} - p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 169 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 461 T^{2} + 94872 T^{4} + 461 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 358 T^{2} - 1643397 T^{4} + 358 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 45 T - 2888 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 13682 T^{2} + 140151243 T^{4} + 13682 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 162 T + 14077 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 144 T - 3653 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 10114 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 9497 T^{2} - 2475533400 T^{4} + 9497 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 100978 T^{2} + 5446452243 T^{4} + 100978 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 97 T - 70098 T^{2} - 97 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 195325 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 414 T + p^{3} T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 + 138274 T^{2} - 23060834565 T^{4} + 138274 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 376 T - 85605 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 600230 T^{2} + 269817670731 T^{4} + 600230 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 588373 T^{2} + 218082503208 T^{4} + 588373 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 592 T + p^{3} T^{2} )^{2}( 1 + 592 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 830 T + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 951730 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 1218094 T^{2} + 986771701875 T^{4} + 1218094 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 1099442 T^{2} + 375800706435 T^{4} + 1099442 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−8.855986920735463874614860258691, −8.519891328251833280702580905558, −8.413185884279875428541249704633, −7.903537479259232717887235571558, −7.84528105952122428444716378950, −7.79028163314352178501885124662, −7.11613072346663574903991750319, −7.06817316862821663289292078068, −6.66436567562850134121176834660, −6.27299138192584091416527866805, −6.03639941165428366765169790036, −5.89824360182748190259681089093, −5.23623047473452404273853853742, −4.87268027808712444603503094980, −4.69886381334394153539946218946, −4.37325363721201500642438261279, −4.25537935098846434846182064415, −3.76226575345647163945360811346, −3.22082876825889628708262070850, −2.92085851538731128315860727926, −2.80015342349965699286892327268, −1.69427528113148729441462086750, −1.36040707242220767582168639413, −1.30707152713293526496204710259, −0.34790610816532466888216770034,
0.34790610816532466888216770034, 1.30707152713293526496204710259, 1.36040707242220767582168639413, 1.69427528113148729441462086750, 2.80015342349965699286892327268, 2.92085851538731128315860727926, 3.22082876825889628708262070850, 3.76226575345647163945360811346, 4.25537935098846434846182064415, 4.37325363721201500642438261279, 4.69886381334394153539946218946, 4.87268027808712444603503094980, 5.23623047473452404273853853742, 5.89824360182748190259681089093, 6.03639941165428366765169790036, 6.27299138192584091416527866805, 6.66436567562850134121176834660, 7.06817316862821663289292078068, 7.11613072346663574903991750319, 7.79028163314352178501885124662, 7.84528105952122428444716378950, 7.903537479259232717887235571558, 8.413185884279875428541249704633, 8.519891328251833280702580905558, 8.855986920735463874614860258691
