| L(s) = 1 | + 10·5-s − 20·9-s − 20·13-s − 84·17-s + 75·25-s + 24·29-s − 364·37-s − 16·41-s − 200·45-s − 652·49-s − 692·53-s + 1.17e3·61-s − 200·65-s − 116·73-s − 329·81-s − 840·85-s + 876·89-s − 2.95e3·97-s − 1.02e3·101-s + 1.50e3·109-s − 3.47e3·113-s + 400·117-s − 2.11e3·121-s + 500·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.740·9-s − 0.426·13-s − 1.19·17-s + 3/5·25-s + 0.153·29-s − 1.61·37-s − 0.0609·41-s − 0.662·45-s − 1.90·49-s − 1.79·53-s + 2.45·61-s − 0.381·65-s − 0.185·73-s − 0.451·81-s − 1.07·85-s + 1.04·89-s − 3.09·97-s − 1.00·101-s + 1.31·109-s − 2.89·113-s + 0.316·117-s − 1.59·121-s + 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 20 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 652 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2118 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 12494 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 21580 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 43126 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 182 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 48548 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 165996 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 346 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 159294 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 586 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 18052 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 666726 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 58 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 542018 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1086420 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 438 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1478 T + p^{3} T^{2} )^{2} \) |
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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
−8.990605580361402488415155311932, −8.831143831680356583622079251309, −8.219576812642405089708596251076, −8.115963879653458301978925709490, −7.40823418545090487208834159434, −6.88822619514463943346314866094, −6.47570820356023366291754443605, −6.44262515441773771627314368472, −5.52063210306383044297655800818, −5.46689028324171985792874291383, −4.87829418835866691937258525964, −4.51835617371829475861424651825, −3.80023968503339439830755981082, −3.34126763632260012801337593641, −2.65606522802566493242444111444, −2.39098853092476686762939464930, −1.70046342961834452284189357302, −1.22392467846219956374302175907, 0, 0,
1.22392467846219956374302175907, 1.70046342961834452284189357302, 2.39098853092476686762939464930, 2.65606522802566493242444111444, 3.34126763632260012801337593641, 3.80023968503339439830755981082, 4.51835617371829475861424651825, 4.87829418835866691937258525964, 5.46689028324171985792874291383, 5.52063210306383044297655800818, 6.44262515441773771627314368472, 6.47570820356023366291754443605, 6.88822619514463943346314866094, 7.40823418545090487208834159434, 8.115963879653458301978925709490, 8.219576812642405089708596251076, 8.831143831680356583622079251309, 8.990605580361402488415155311932
