L(s) = 1 | + 10·3-s + 26·4-s + 10·5-s + 70·7-s − 87·9-s + 178·11-s + 260·12-s − 10·13-s + 100·15-s + 420·16-s − 970·17-s + 260·20-s + 700·21-s − 525·25-s − 1.93e3·27-s + 1.82e3·28-s + 382·29-s + 1.78e3·33-s + 700·35-s − 2.26e3·36-s − 100·39-s + 4.62e3·44-s − 870·45-s − 4.39e3·47-s + 4.20e3·48-s + 2.49e3·49-s − 9.70e3·51-s + ⋯ |
L(s) = 1 | + 10/9·3-s + 13/8·4-s + 2/5·5-s + 10/7·7-s − 1.07·9-s + 1.47·11-s + 1.80·12-s − 0.0591·13-s + 4/9·15-s + 1.64·16-s − 3.35·17-s + 0.649·20-s + 1.58·21-s − 0.839·25-s − 2.64·27-s + 2.32·28-s + 0.454·29-s + 1.63·33-s + 4/7·35-s − 1.74·36-s − 0.0657·39-s + 2.39·44-s − 0.429·45-s − 1.98·47-s + 1.82·48-s + 1.04·49-s − 3.72·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(3.883736733\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.883736733\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | $C_2$ | \( 1 - 2 p T + p^{4} T^{2} \) |
| 7 | $C_2$ | \( 1 - 10 p T + p^{4} T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - 13 p T^{2} + p^{8} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 5 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 89 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 485 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 212042 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 68906 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 191 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 737642 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 794 p^{2} T^{2} + p^{8} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2845078 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6695306 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2195 T + p^{4} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 13261538 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 11092322 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 23947082 T^{2} + p^{8} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 36108866 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4454 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8650 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5561 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 1990 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 124831082 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 9235 T + p^{4} T^{2} )^{2} \) |
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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
−15.78605964050010973043571338674, −15.36150055342789548375809918933, −14.85273661187535033468194370028, −14.45065008951463534921524175665, −13.82124915359663082856912551027, −13.42837995572760950408359374228, −12.20360554861326034382482696112, −11.51230780852540892505119352806, −11.15109951002768807288444074118, −11.00590441988255020458470577646, −9.558580304904769233577668359034, −8.905162000563469516246488216410, −8.372013421659663015342318746996, −7.73872135544898343163271737015, −6.49968452293574160235912884544, −6.41831416280636474931610874263, −4.96861067944583778768211865705, −3.67335978508331130762491268181, −2.21157011314326723139645887774, −2.07092370770431135743982480020,
2.07092370770431135743982480020, 2.21157011314326723139645887774, 3.67335978508331130762491268181, 4.96861067944583778768211865705, 6.41831416280636474931610874263, 6.49968452293574160235912884544, 7.73872135544898343163271737015, 8.372013421659663015342318746996, 8.905162000563469516246488216410, 9.558580304904769233577668359034, 11.00590441988255020458470577646, 11.15109951002768807288444074118, 11.51230780852540892505119352806, 12.20360554861326034382482696112, 13.42837995572760950408359374228, 13.82124915359663082856912551027, 14.45065008951463534921524175665, 14.85273661187535033468194370028, 15.36150055342789548375809918933, 15.78605964050010973043571338674