| L(s) = 1 | − 60·11-s − 56·31-s − 156·41-s + 368·61-s − 408·71-s + 81·81-s − 96·101-s + 1.76e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 5.45·11-s − 1.80·31-s − 3.80·41-s + 6.03·61-s − 5.74·71-s + 81-s − 0.950·101-s + 14.5·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.05991266295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05991266295\) |
| \(L(2)\) |
|
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 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 2302 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 15 T + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 49154 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 136559 T^{4} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 433 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 342818 T^{4} + p^{8} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2}( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 2728222 T^{4} + p^{8} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 39 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 597598 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 8995486 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 5984062 T^{4} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 6926 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 92 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 1364783 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 102 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 56221679 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 1666 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 10000417 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 8273 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - 174838462 T^{4} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.12663606521312104299527401487, −10.03200107014382119866643329313, −9.497891816238118008906101293823, −9.194116320369010508571124158849, −8.532391733368428936304058810276, −8.339446672242895815054326697900, −8.319824931294298850157859651179, −8.165761712634884059439460832122, −7.50349323335030245185083330513, −7.34661908671223330660857759975, −7.13995274658889710804397063313, −6.97450910128623942551330582241, −6.09470816752731902514474223856, −5.93757779498629995785410231029, −5.38411535821729750399263491258, −5.15583856402041304581311932571, −5.14629085487707576902164514812, −4.90690938396703502495608259492, −4.09759739536253158914106714044, −3.57808428532582540422917974780, −3.12340895329761995479241918007, −2.63326025033742628133466937747, −2.42746653859259225954538169066, −1.81504161112618167300818922635, −0.10696868760892397094559084782,
0.10696868760892397094559084782, 1.81504161112618167300818922635, 2.42746653859259225954538169066, 2.63326025033742628133466937747, 3.12340895329761995479241918007, 3.57808428532582540422917974780, 4.09759739536253158914106714044, 4.90690938396703502495608259492, 5.14629085487707576902164514812, 5.15583856402041304581311932571, 5.38411535821729750399263491258, 5.93757779498629995785410231029, 6.09470816752731902514474223856, 6.97450910128623942551330582241, 7.13995274658889710804397063313, 7.34661908671223330660857759975, 7.50349323335030245185083330513, 8.165761712634884059439460832122, 8.319824931294298850157859651179, 8.339446672242895815054326697900, 8.532391733368428936304058810276, 9.194116320369010508571124158849, 9.497891816238118008906101293823, 10.03200107014382119866643329313, 10.12663606521312104299527401487
