| L(s) = 1 | + 3-s + 2·7-s + 9-s + 12·13-s + 8·19-s + 2·21-s − 6·25-s + 27-s − 20·37-s + 12·39-s + 8·43-s + 3·49-s + 8·57-s + 12·61-s + 2·63-s − 8·67-s + 20·73-s − 6·75-s + 81-s + 24·91-s − 28·97-s − 16·103-s − 4·109-s − 20·111-s + 12·117-s − 6·121-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 3.32·13-s + 1.83·19-s + 0.436·21-s − 6/5·25-s + 0.192·27-s − 3.28·37-s + 1.92·39-s + 1.21·43-s + 3/7·49-s + 1.05·57-s + 1.53·61-s + 0.251·63-s − 0.977·67-s + 2.34·73-s − 0.692·75-s + 1/9·81-s + 2.51·91-s − 2.84·97-s − 1.57·103-s − 0.383·109-s − 1.89·111-s + 1.10·117-s − 0.545·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.164303633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.164303633\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.579197535720292899748335278003, −8.427817591055194605299416158333, −7.963562405403294064353399542187, −7.46998199559376075566652373057, −6.89499765960031010222506198709, −6.48878700424638503015501865218, −5.69761771888975023011122238972, −5.58110052520146908580921209772, −4.96240850233932644050318806427, −4.04752481141371955267217260485, −3.57539886636491016875562737772, −3.54052974946578322805924483948, −2.47676327970158091646809943718, −1.50800619952171259085588630937, −1.22168166913217334012046287130,
1.22168166913217334012046287130, 1.50800619952171259085588630937, 2.47676327970158091646809943718, 3.54052974946578322805924483948, 3.57539886636491016875562737772, 4.04752481141371955267217260485, 4.96240850233932644050318806427, 5.58110052520146908580921209772, 5.69761771888975023011122238972, 6.48878700424638503015501865218, 6.89499765960031010222506198709, 7.46998199559376075566652373057, 7.963562405403294064353399542187, 8.427817591055194605299416158333, 8.579197535720292899748335278003
