| L(s) = 1 | + 4-s − 4·7-s − 3·9-s − 3·16-s − 12·17-s − 4·28-s − 3·36-s + 4·37-s + 12·41-s + 16·43-s + 24·47-s + 9·49-s − 24·59-s + 12·63-s − 7·64-s − 16·67-s − 12·68-s + 16·79-s + 9·81-s + 12·89-s − 12·101-s − 4·109-s + 12·112-s + 48·119-s + 10·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.51·7-s − 9-s − 3/4·16-s − 2.91·17-s − 0.755·28-s − 1/2·36-s + 0.657·37-s + 1.87·41-s + 2.43·43-s + 3.50·47-s + 9/7·49-s − 3.12·59-s + 1.51·63-s − 7/8·64-s − 1.95·67-s − 1.45·68-s + 1.80·79-s + 81-s + 1.27·89-s − 1.19·101-s − 0.383·109-s + 1.13·112-s + 4.40·119-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8661560179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8661560179\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−11.05728907308294597640692751467, −10.64505167473116839181775573867, −10.55601873144198600642825540133, −9.393858134678248784463624231204, −9.291608285219077657063013782677, −8.966822830681800343960945127234, −8.776884818319404234225133865568, −7.65404072485387587447613205576, −7.56293213253545126169148953859, −6.91856009216061670547587563037, −6.39309111447211168970554059232, −5.99747198042375773798681745579, −5.95496263296676107145589615451, −4.88773036380882085998920885317, −4.14485422861587002751360888280, −4.10825837911786170154677775303, −2.87131288274295921635037414432, −2.64352353327532264001546469447, −2.17070756404279439413224669342, −0.51127384343248679183538932567,
0.51127384343248679183538932567, 2.17070756404279439413224669342, 2.64352353327532264001546469447, 2.87131288274295921635037414432, 4.10825837911786170154677775303, 4.14485422861587002751360888280, 4.88773036380882085998920885317, 5.95496263296676107145589615451, 5.99747198042375773798681745579, 6.39309111447211168970554059232, 6.91856009216061670547587563037, 7.56293213253545126169148953859, 7.65404072485387587447613205576, 8.776884818319404234225133865568, 8.966822830681800343960945127234, 9.291608285219077657063013782677, 9.393858134678248784463624231204, 10.55601873144198600642825540133, 10.64505167473116839181775573867, 11.05728907308294597640692751467
