| L(s) = 1 | + 47·2-s + 4.09e3·4-s + 4.73e5·8-s − 5.31e5·9-s − 3.06e5·11-s + 2.22e7·16-s − 2.49e7·18-s − 1.43e7·22-s − 2.20e8·23-s − 2.44e8·25-s − 1.47e9·29-s + 1.94e9·32-s − 2.17e9·36-s − 5.10e9·37-s + 6.77e9·43-s − 1.25e9·44-s − 1.03e10·46-s − 1.14e10·50-s + 4.17e10·53-s − 6.94e10·58-s + 1.55e11·64-s + 1.78e11·67-s − 3.94e11·71-s − 2.51e11·72-s − 2.40e11·74-s − 3.77e11·79-s + 3.18e11·86-s + ⋯ |
| L(s) = 1 | + 0.734·2-s + 4-s + 1.80·8-s − 9-s − 0.172·11-s + 1.32·16-s − 0.734·18-s − 0.126·22-s − 1.49·23-s − 25-s − 2.48·29-s + 1.80·32-s − 36-s − 1.99·37-s + 1.07·43-s − 0.172·44-s − 1.09·46-s − 0.734·50-s + 1.88·53-s − 1.82·58-s + 2.26·64-s + 1.96·67-s − 3.08·71-s − 1.80·72-s − 1.46·74-s − 1.55·79-s + 0.787·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+6)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(2.848036092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.848036092\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 47 T - 1887 T^{2} - 47 p^{12} T^{3} + p^{24} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p^{6} T + p^{12} T^{2} )( 1 + p^{6} T + p^{12} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p^{6} T + p^{12} T^{2} )( 1 + p^{6} T + p^{12} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 306322 T - 3044595209037 T^{2} + 306322 p^{12} T^{3} + p^{24} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{6} T + p^{12} T^{2} )( 1 + p^{6} T + p^{12} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p^{6} T + p^{12} T^{2} )( 1 + p^{6} T + p^{12} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 220762978 T + 26821668023408163 T^{2} + 220762978 p^{12} T^{3} + p^{24} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 739273358 T + p^{12} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{6} T + p^{12} T^{2} )( 1 + p^{6} T + p^{12} T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 5108772818 T + 19516607700095625843 T^{2} + 5108772818 p^{12} T^{3} + p^{24} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 3388378898 T + p^{12} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{6} T + p^{12} T^{2} )( 1 + p^{6} T + p^{12} T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 41794002542 T + \)\(12\!\cdots\!23\)\( T^{2} - 41794002542 p^{12} T^{3} + p^{24} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p^{6} T + p^{12} T^{2} )( 1 + p^{6} T + p^{12} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{6} T + p^{12} T^{2} )( 1 + p^{6} T + p^{12} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 178008750862 T + \)\(23\!\cdots\!83\)\( T^{2} - 178008750862 p^{12} T^{3} + p^{24} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 197404987358 T + p^{12} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p^{6} T + p^{12} T^{2} )( 1 + p^{6} T + p^{12} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 377568555842 T + \)\(83\!\cdots\!23\)\( T^{2} + 377568555842 p^{12} T^{3} + p^{24} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{6} T + p^{12} T^{2} )( 1 + p^{6} T + p^{12} T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.29415718882207784210037614211, −12.86037860761515171903996088883, −12.01706736459374302029502489643, −11.55258284674924817352678669843, −11.20221030411514774782385017987, −10.36800081636220610023346636642, −10.09211318350927871020113461610, −9.055055110771186676760513067854, −8.372965353900674601182269630466, −7.52539384789293191843927933512, −7.35177387940208895928401809053, −6.35763627365760385230584781800, −5.59283847168904871466699857754, −5.36999510606636839844458581670, −4.13530340770100959489126612067, −3.83884036748593142617006882887, −2.83058986954250835646532312112, −2.00798465308089808033197082358, −1.66973886941159877383069029083, −0.37560786054807558751925049722,
0.37560786054807558751925049722, 1.66973886941159877383069029083, 2.00798465308089808033197082358, 2.83058986954250835646532312112, 3.83884036748593142617006882887, 4.13530340770100959489126612067, 5.36999510606636839844458581670, 5.59283847168904871466699857754, 6.35763627365760385230584781800, 7.35177387940208895928401809053, 7.52539384789293191843927933512, 8.372965353900674601182269630466, 9.055055110771186676760513067854, 10.09211318350927871020113461610, 10.36800081636220610023346636642, 11.20221030411514774782385017987, 11.55258284674924817352678669843, 12.01706736459374302029502489643, 12.86037860761515171903996088883, 13.29415718882207784210037614211
