| L(s) = 1 | − 8·4-s − 9·9-s + 48·16-s + 74·19-s − 26·31-s + 72·36-s − 23·49-s + 94·61-s − 256·64-s − 592·76-s + 284·79-s + 81·81-s − 286·109-s + 242·121-s + 208·124-s + 127-s + 131-s + 137-s + 139-s − 432·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 337·169-s − 666·171-s + ⋯ |
| L(s) = 1 | − 2·4-s − 9-s + 3·16-s + 3.89·19-s − 0.838·31-s + 2·36-s − 0.469·49-s + 1.54·61-s − 4·64-s − 7.78·76-s + 3.59·79-s + 81-s − 2.62·109-s + 2·121-s + 1.67·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 3·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.99·169-s − 3.89·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8150312228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8150312228\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 23 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 337 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2062 T^{2} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 23 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 2903 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 8542 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9743 T^{2} + p^{4} T^{4} \) |
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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
−14.72730958661177218596959692707, −13.76713982750329349389723601579, −13.76165706987377172305799302461, −13.17807981962901799163044293868, −12.37421305010241471478071671033, −11.95124334429892887341315238770, −11.39499158191839761953432580722, −10.58389939498170442206218058872, −9.788244000507150098621056394312, −9.341114723153192091921942940831, −9.203868736757563951448684387432, −8.168283394102852171471206320964, −7.929308817635791345093554598675, −7.09129825850865438987549049126, −5.75462090382639367735554950455, −5.35047507057977091293151893717, −4.89295320974082660446728005919, −3.66397516908791709869342815243, −3.20203463841436982829307878115, −0.859229389880520879695665431533,
0.859229389880520879695665431533, 3.20203463841436982829307878115, 3.66397516908791709869342815243, 4.89295320974082660446728005919, 5.35047507057977091293151893717, 5.75462090382639367735554950455, 7.09129825850865438987549049126, 7.929308817635791345093554598675, 8.168283394102852171471206320964, 9.203868736757563951448684387432, 9.341114723153192091921942940831, 9.788244000507150098621056394312, 10.58389939498170442206218058872, 11.39499158191839761953432580722, 11.95124334429892887341315238770, 12.37421305010241471478071671033, 13.17807981962901799163044293868, 13.76165706987377172305799302461, 13.76713982750329349389723601579, 14.72730958661177218596959692707
