| L(s) = 1 | − 3-s + 8·7-s + 9-s + 4·13-s + 8·19-s − 8·21-s + 25-s − 27-s − 16·31-s + 4·37-s − 4·39-s + 8·43-s + 34·49-s − 8·57-s − 20·61-s + 8·63-s + 8·67-s + 4·73-s − 75-s − 16·79-s + 81-s + 32·91-s + 16·93-s + 4·97-s + 8·103-s − 20·109-s − 4·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 3.02·7-s + 1/3·9-s + 1.10·13-s + 1.83·19-s − 1.74·21-s + 1/5·25-s − 0.192·27-s − 2.87·31-s + 0.657·37-s − 0.640·39-s + 1.21·43-s + 34/7·49-s − 1.05·57-s − 2.56·61-s + 1.00·63-s + 0.977·67-s + 0.468·73-s − 0.115·75-s − 1.80·79-s + 1/9·81-s + 3.35·91-s + 1.65·93-s + 0.406·97-s + 0.788·103-s − 1.91·109-s − 0.379·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.241776282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.241776282\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.206120678796429021467253404807, −8.713350404322888410914105544855, −7.965215938754459889106897012614, −7.88297907167905508023143292047, −7.38482725826507043429631860666, −6.96170559755127999142057961261, −5.85236328773314391596499497567, −5.76555679083117871916592673419, −4.97026786036012165020386374880, −4.95303891523898796353167575120, −4.08471431361169401081275954745, −3.62987230449253956202768748221, −2.50374962358163435732612030839, −1.44399343799308961081305068602, −1.36243712364701817507861978801,
1.36243712364701817507861978801, 1.44399343799308961081305068602, 2.50374962358163435732612030839, 3.62987230449253956202768748221, 4.08471431361169401081275954745, 4.95303891523898796353167575120, 4.97026786036012165020386374880, 5.76555679083117871916592673419, 5.85236328773314391596499497567, 6.96170559755127999142057961261, 7.38482725826507043429631860666, 7.88297907167905508023143292047, 7.965215938754459889106897012614, 8.713350404322888410914105544855, 9.206120678796429021467253404807
