| L(s) = 1 | − 16·7-s + 26·9-s + 28·17-s − 304·23-s − 448·31-s − 140·41-s + 672·47-s − 494·49-s − 416·63-s + 144·71-s + 588·73-s + 928·79-s − 53·81-s + 532·89-s − 1.98e3·97-s + 2.35e3·103-s + 3.42e3·113-s − 448·119-s + 2.41e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 728·153-s + 157-s + ⋯ |
| L(s) = 1 | − 0.863·7-s + 0.962·9-s + 0.399·17-s − 2.75·23-s − 2.59·31-s − 0.533·41-s + 2.08·47-s − 1.44·49-s − 0.831·63-s + 0.240·71-s + 0.942·73-s + 1.32·79-s − 0.0727·81-s + 0.633·89-s − 2.08·97-s + 2.24·103-s + 2.84·113-s − 0.345·119-s + 1.81·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.384·153-s + 0.000508·157-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
| \(L(2)\) |
\(\approx\) |
\(0.8132408779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8132408779\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 26 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2410 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1594 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12346 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 152 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 23578 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 42058 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 33878 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 336 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 296746 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 125130 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 444890 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 571034 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 294 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 464 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 846522 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 266 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 994 T + p^{3} T^{2} )^{2} \) |
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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
−9.945180541227773339756653636036, −9.898476345477803808246889285268, −9.157821486564181833801476207821, −9.093052370770075259444537916697, −8.223625288263474519782614322533, −8.026227797089654370630206593256, −7.34953398701924564583947121570, −7.22270888911280730521447734003, −6.63231441976888985963369956135, −5.96289589358899119272046804422, −5.93434359289415222397581019190, −5.24718275168003991439486415249, −4.62120747926129853004260095382, −4.08533632653934119034433227853, −3.49601985253720186966975684082, −3.46771193590558800220132001357, −2.20252078332611003468659867997, −2.05035848570045410945505335531, −1.20275084940894231248047869013, −0.24853484667633110509320176956,
0.24853484667633110509320176956, 1.20275084940894231248047869013, 2.05035848570045410945505335531, 2.20252078332611003468659867997, 3.46771193590558800220132001357, 3.49601985253720186966975684082, 4.08533632653934119034433227853, 4.62120747926129853004260095382, 5.24718275168003991439486415249, 5.93434359289415222397581019190, 5.96289589358899119272046804422, 6.63231441976888985963369956135, 7.22270888911280730521447734003, 7.34953398701924564583947121570, 8.026227797089654370630206593256, 8.223625288263474519782614322533, 9.093052370770075259444537916697, 9.157821486564181833801476207821, 9.898476345477803808246889285268, 9.945180541227773339756653636036
