| L(s) = 1 | − 110·5-s + 290·9-s − 296·11-s − 4.44e3·19-s + 8.97e3·25-s − 540·29-s + 4.09e3·31-s − 4.79e3·41-s − 3.19e4·45-s + 8.65e3·49-s + 3.25e4·55-s + 7.94e4·59-s + 8.45e4·61-s + 8.49e3·71-s + 7.05e4·79-s + 2.50e4·81-s + 1.70e5·89-s + 4.88e5·95-s − 8.58e4·99-s + 8.59e3·101-s − 7.19e4·109-s − 2.56e5·121-s − 6.43e5·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.96·5-s + 1.19·9-s − 0.737·11-s − 2.82·19-s + 2.87·25-s − 0.119·29-s + 0.765·31-s − 0.445·41-s − 2.34·45-s + 0.514·49-s + 1.45·55-s + 2.97·59-s + 2.91·61-s + 0.200·71-s + 1.27·79-s + 0.424·81-s + 2.28·89-s + 5.55·95-s − 0.880·99-s + 0.0838·101-s − 0.580·109-s − 1.59·121-s − 3.68·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.433313809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.433313809\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 22 p T + p^{5} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 290 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 8650 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 148 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 274730 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1354590 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2220 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11320170 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 270 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2048 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 119573530 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2398 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 288754450 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 344584890 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 827605690 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 39740 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 42298 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1669968610 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4248 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3239892370 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 35280 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7103795010 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 85210 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 7720618690 T^{2} + p^{10} T^{4} \) |
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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
−10.80512971703842205938793926893, −10.78438816917617883692052374153, −10.20389991219384000273201360489, −9.821021895269934300427676916367, −8.905186250823924820764183138716, −8.554766010370562719683013998795, −8.112614617917449204419180571065, −7.895379098747612100151775831001, −7.01254228920753954371332083765, −6.93452499489299452121806688560, −6.42339261506598745320188060853, −5.47686317546994207955442921791, −4.79269969644412968960353032309, −4.41709671029085249603609144419, −3.86324907561574315849105982650, −3.60940092228080351029101369613, −2.51131129197342711917014645541, −2.07406856841339630670662715508, −0.860149954218277697006783331640, −0.42311220523008900598544120931,
0.42311220523008900598544120931, 0.860149954218277697006783331640, 2.07406856841339630670662715508, 2.51131129197342711917014645541, 3.60940092228080351029101369613, 3.86324907561574315849105982650, 4.41709671029085249603609144419, 4.79269969644412968960353032309, 5.47686317546994207955442921791, 6.42339261506598745320188060853, 6.93452499489299452121806688560, 7.01254228920753954371332083765, 7.895379098747612100151775831001, 8.112614617917449204419180571065, 8.554766010370562719683013998795, 8.905186250823924820764183138716, 9.821021895269934300427676916367, 10.20389991219384000273201360489, 10.78438816917617883692052374153, 10.80512971703842205938793926893
