| L(s) = 1 | + 2·3-s − 3·4-s − 5-s + 3·9-s − 6·12-s − 2·15-s + 5·16-s − 2·17-s + 8·19-s + 3·20-s + 25-s + 4·27-s − 9·36-s + 20·37-s − 3·45-s + 10·48-s − 14·49-s − 4·51-s + 16·57-s − 8·59-s + 6·60-s − 3·64-s + 6·68-s − 20·73-s + 2·75-s − 24·76-s − 5·80-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s − 0.447·5-s + 9-s − 1.73·12-s − 0.516·15-s + 5/4·16-s − 0.485·17-s + 1.83·19-s + 0.670·20-s + 1/5·25-s + 0.769·27-s − 3/2·36-s + 3.28·37-s − 0.447·45-s + 1.44·48-s − 2·49-s − 0.560·51-s + 2.11·57-s − 1.04·59-s + 0.774·60-s − 3/8·64-s + 0.727·68-s − 2.34·73-s + 0.230·75-s − 2.75·76-s − 0.559·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.702200269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.702200269\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.843956595114680827452383410006, −8.324027666892170998148797227210, −7.88272580417467619012367098000, −7.68003844385765985083943135134, −7.16750910474070634391114159922, −6.44628802162953221998351414777, −5.86497653842965752491328535183, −5.26301391890678078722159871955, −4.52160444884874669934496061646, −4.47629935336188211298931611555, −3.84194557004788343759428710030, −3.01549784140686353642765162463, −2.95453421762719696432427292949, −1.69452749631301531601828409816, −0.75602502315476554931106499220,
0.75602502315476554931106499220, 1.69452749631301531601828409816, 2.95453421762719696432427292949, 3.01549784140686353642765162463, 3.84194557004788343759428710030, 4.47629935336188211298931611555, 4.52160444884874669934496061646, 5.26301391890678078722159871955, 5.86497653842965752491328535183, 6.44628802162953221998351414777, 7.16750910474070634391114159922, 7.68003844385765985083943135134, 7.88272580417467619012367098000, 8.324027666892170998148797227210, 8.843956595114680827452383410006
