| L(s) = 1 | + 8·3-s − 8·7-s − 6·9-s − 64·21-s + 104·23-s − 392·27-s − 316·29-s − 340·41-s − 632·43-s − 488·47-s − 638·49-s + 164·61-s + 48·63-s − 1.38e3·67-s + 832·69-s − 1.71e3·81-s − 1.88e3·83-s − 2.52e3·87-s + 12·89-s + 2.02e3·101-s − 88·103-s − 3.62e3·107-s + 4.02e3·109-s − 806·121-s − 2.72e3·123-s + 127-s − 5.05e3·129-s + ⋯ |
| L(s) = 1 | + 1.53·3-s − 0.431·7-s − 2/9·9-s − 0.665·21-s + 0.942·23-s − 2.79·27-s − 2.02·29-s − 1.29·41-s − 2.24·43-s − 1.51·47-s − 1.86·49-s + 0.344·61-s + 0.0959·63-s − 2.52·67-s + 1.45·69-s − 2.35·81-s − 2.48·83-s − 3.11·87-s + 0.0142·89-s + 1.99·101-s − 0.0841·103-s − 3.27·107-s + 3.53·109-s − 0.605·121-s − 1.99·123-s + 0.000698·127-s − 3.45·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 806 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 3930 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7970 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2986 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 52 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 158 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 29886 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22890 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 170 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 316 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 244 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 52298 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6842 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 82 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 692 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 182482 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 592434 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 867294 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 940 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 665346 T^{2} + p^{6} T^{4} \) |
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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
−9.400097866505981300357598014738, −9.287240871899729315641205513772, −8.685937561397672258895861497817, −8.574921341532598736542671858358, −7.926260230411318492674464067068, −7.81057221619692224516987487449, −7.10189904281048140512621077380, −6.76414057158544763348522265944, −6.14639696963288802742516766422, −5.71314752121280629264017633902, −5.12788320388749020751040359257, −4.75619992829946513282469501778, −3.82351669212760419386627883124, −3.49912174229817472368492834088, −3.02036719171712036895509176125, −2.79526411175839626968638715703, −1.73716366806993572092927253180, −1.68200324377131066143234063308, 0, 0,
1.68200324377131066143234063308, 1.73716366806993572092927253180, 2.79526411175839626968638715703, 3.02036719171712036895509176125, 3.49912174229817472368492834088, 3.82351669212760419386627883124, 4.75619992829946513282469501778, 5.12788320388749020751040359257, 5.71314752121280629264017633902, 6.14639696963288802742516766422, 6.76414057158544763348522265944, 7.10189904281048140512621077380, 7.81057221619692224516987487449, 7.926260230411318492674464067068, 8.574921341532598736542671858358, 8.685937561397672258895861497817, 9.287240871899729315641205513772, 9.400097866505981300357598014738
