| L(s) = 1 | − 84·5-s − 27·9-s − 364·13-s − 492·17-s + 4.04e3·25-s + 156·29-s + 1.06e3·37-s − 1.83e3·41-s + 2.26e3·45-s − 1.00e3·49-s − 9.25e3·53-s + 2.69e3·61-s + 3.05e4·65-s − 1.85e3·73-s + 729·81-s + 4.13e4·85-s + 2.31e4·89-s − 2.62e4·97-s − 1.09e4·101-s − 3.23e4·109-s + 3.68e3·113-s + 9.82e3·117-s + 2.88e4·121-s − 1.38e5·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3.35·5-s − 1/3·9-s − 2.15·13-s − 1.70·17-s + 6.46·25-s + 0.185·29-s + 0.774·37-s − 1.09·41-s + 1.11·45-s − 0.418·49-s − 3.29·53-s + 0.723·61-s + 7.23·65-s − 0.347·73-s + 1/9·81-s + 5.72·85-s + 2.92·89-s − 2.78·97-s − 1.07·101-s − 2.72·109-s + 0.288·113-s + 0.717·117-s + 1.97·121-s − 8.88·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.04244202995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04244202995\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 42 T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 1006 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 28850 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 14 p T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 246 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 246770 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 190 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 78 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 330670 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 530 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 918 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6111410 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4550206 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4626 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 24182450 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1346 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 39119090 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 47477954 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 926 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 58545362 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 48908686 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 11586 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13118 T + p^{4} T^{2} )^{2} \) |
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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
−15.37583898284340874073567323191, −14.69813388049315739565035186947, −14.51395582617085866355526371091, −13.31788979148204115279696810690, −12.57659196562779654766983422145, −12.11788520647609452952830611026, −11.82252072088480578314322528708, −11.06364021271221638172635984402, −10.96605336005405003038884619459, −9.737689325801028331617424946321, −8.949942189045713755909385347229, −8.040285422325152285462884953920, −7.969066972312479192441174093380, −7.13957588837042510243482604504, −6.68722556865995270354279333798, −4.80562169706109947135752656155, −4.59785358008756302741252094972, −3.67978856994224562271665549233, −2.74842984673848806790974174259, −0.12304026133366934218400926097,
0.12304026133366934218400926097, 2.74842984673848806790974174259, 3.67978856994224562271665549233, 4.59785358008756302741252094972, 4.80562169706109947135752656155, 6.68722556865995270354279333798, 7.13957588837042510243482604504, 7.969066972312479192441174093380, 8.040285422325152285462884953920, 8.949942189045713755909385347229, 9.737689325801028331617424946321, 10.96605336005405003038884619459, 11.06364021271221638172635984402, 11.82252072088480578314322528708, 12.11788520647609452952830611026, 12.57659196562779654766983422145, 13.31788979148204115279696810690, 14.51395582617085866355526371091, 14.69813388049315739565035186947, 15.37583898284340874073567323191
