| L(s) = 1 | + 8·3-s − 38·5-s − 401·9-s − 424·11-s − 924·13-s − 304·15-s − 2.34e3·17-s + 360·19-s + 12·23-s − 1.46e3·25-s − 4.98e3·27-s − 7.05e3·29-s − 3.54e3·31-s − 3.39e3·33-s + 1.10e4·37-s − 7.39e3·39-s + 3.50e3·41-s + 1.26e4·43-s + 1.52e4·45-s + 2.29e4·47-s − 1.87e4·51-s + 3.04e3·53-s + 1.61e4·55-s + 2.88e3·57-s + 6.58e4·59-s − 4.24e4·61-s + 3.51e4·65-s + ⋯ |
| L(s) = 1 | + 0.513·3-s − 0.679·5-s − 1.65·9-s − 1.05·11-s − 1.51·13-s − 0.348·15-s − 1.96·17-s + 0.228·19-s + 0.00473·23-s − 0.469·25-s − 1.31·27-s − 1.55·29-s − 0.663·31-s − 0.542·33-s + 1.33·37-s − 0.778·39-s + 0.325·41-s + 1.04·43-s + 1.12·45-s + 1.51·47-s − 1.01·51-s + 0.148·53-s + 0.718·55-s + 0.117·57-s + 2.46·59-s − 1.46·61-s + 1.03·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9397006712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9397006712\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 8 T + 155 p T^{2} - 8 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 38 T + 2911 T^{2} + 38 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 424 T + 347473 T^{2} + 424 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 924 T + 927022 T^{2} + 924 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 138 p T + 3570955 T^{2} + 138 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 360 T + 2145625 T^{2} - 360 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 12 T + 12696565 T^{2} - 12 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7052 T + 35324974 T^{2} + 7052 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3548 T + 41490053 T^{2} + 3548 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11090 T + 146343239 T^{2} - 11090 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3500 T + 206898214 T^{2} - 3500 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12680 T + 267378054 T^{2} - 12680 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 22956 T + 491525173 T^{2} - 22956 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3042 T + 716414839 T^{2} - 3042 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 65808 T + 2502089257 T^{2} - 65808 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 42486 T + 1501996159 T^{2} + 42486 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 42312 T + 3116577793 T^{2} + 42312 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2208 T + 3433192846 T^{2} - 2208 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 50506 T + 2773040987 T^{2} + 50506 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 9004 T + 5176592589 T^{2} + 9004 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 104328 T + 9837878230 T^{2} - 104328 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 26666 T + 8107102555 T^{2} + 26666 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2156 p T + 28107307478 T^{2} - 2156 p^{6} T^{3} + p^{10} T^{4} \) |
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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
−9.438242398757372951342394793622, −9.307759976446065559328091921899, −8.911036664089897173751317891515, −8.564817453295444105363838353724, −7.82165641951233527777299216094, −7.77485717529361306598533962398, −7.35001764832547198476923126388, −6.92155500600726240661801680666, −6.04555844584081343154050568686, −5.86685153999155970580411849245, −5.29352396206935000563783443268, −4.88358503213123938750922381228, −4.17588667962140051419770661549, −3.96323708230849900041991978065, −3.08286700135097381506353677487, −2.75779367214443418436288749934, −2.14233959711887185248407727140, −2.09630529448598168513959063168, −0.54699455183564451493520991513, −0.32223496566032188068650867867,
0.32223496566032188068650867867, 0.54699455183564451493520991513, 2.09630529448598168513959063168, 2.14233959711887185248407727140, 2.75779367214443418436288749934, 3.08286700135097381506353677487, 3.96323708230849900041991978065, 4.17588667962140051419770661549, 4.88358503213123938750922381228, 5.29352396206935000563783443268, 5.86685153999155970580411849245, 6.04555844584081343154050568686, 6.92155500600726240661801680666, 7.35001764832547198476923126388, 7.77485717529361306598533962398, 7.82165641951233527777299216094, 8.564817453295444105363838353724, 8.911036664089897173751317891515, 9.307759976446065559328091921899, 9.438242398757372951342394793622
