| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8·7-s + 8-s + 9-s + 10-s − 12-s + 8·14-s − 15-s + 16-s − 12·17-s + 18-s + 20-s − 8·21-s − 24-s + 25-s − 27-s + 8·28-s − 30-s + 32-s − 12·34-s + 8·35-s + 36-s + 40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 3.02·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 2.13·14-s − 0.258·15-s + 1/4·16-s − 2.91·17-s + 0.235·18-s + 0.223·20-s − 1.74·21-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 1.51·28-s − 0.182·30-s + 0.176·32-s − 2.05·34-s + 1.35·35-s + 1/6·36-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.901008377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.901008377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.415401762585890013045544643092, −8.749862280172923617009160642326, −8.713350404322888410914105544855, −7.88297907167905508023143292047, −7.56689807773085737112376729195, −6.96170559755127999142057961261, −6.38949936725272530282824625186, −5.85236328773314391596499497567, −5.07079211702149946545201877286, −4.97026786036012165020386374880, −4.30912662284802770612117339996, −4.08471431361169401081275954745, −2.50374962358163435732612030839, −2.07452981833800877942993394416, −1.36243712364701817507861978801,
1.36243712364701817507861978801, 2.07452981833800877942993394416, 2.50374962358163435732612030839, 4.08471431361169401081275954745, 4.30912662284802770612117339996, 4.97026786036012165020386374880, 5.07079211702149946545201877286, 5.85236328773314391596499497567, 6.38949936725272530282824625186, 6.96170559755127999142057961261, 7.56689807773085737112376729195, 7.88297907167905508023143292047, 8.713350404322888410914105544855, 8.749862280172923617009160642326, 9.415401762585890013045544643092
