| L(s) = 1 | − 3·9-s + 8·11-s + 4·13-s − 6·25-s − 4·37-s + 16·47-s + 49-s − 12·61-s + 16·71-s + 20·73-s + 9·81-s − 16·83-s − 12·97-s − 24·99-s + 24·107-s − 20·109-s − 12·117-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 9-s + 2.41·11-s + 1.10·13-s − 6/5·25-s − 0.657·37-s + 2.33·47-s + 1/7·49-s − 1.53·61-s + 1.89·71-s + 2.34·73-s + 81-s − 1.75·83-s − 1.21·97-s − 2.41·99-s + 2.32·107-s − 1.91·109-s − 1.10·117-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.522064778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.522064778\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.886707849578587066687733658491, −9.256691510852288615675806117776, −9.117894034638208548514808203742, −8.560607806232820418165457845354, −8.093537689649372229665039418958, −7.40397429472305265033368100557, −6.67949194679031572918581545842, −6.35373904523042773985731435784, −5.82613210142107824134972732669, −5.29340219049011126125230323142, −4.18465932715971087267449515138, −3.89550776474805329328185747620, −3.26735667825841356907489548421, −2.14556791802447766709825303506, −1.15037892508990574535308118020,
1.15037892508990574535308118020, 2.14556791802447766709825303506, 3.26735667825841356907489548421, 3.89550776474805329328185747620, 4.18465932715971087267449515138, 5.29340219049011126125230323142, 5.82613210142107824134972732669, 6.35373904523042773985731435784, 6.67949194679031572918581545842, 7.40397429472305265033368100557, 8.093537689649372229665039418958, 8.560607806232820418165457845354, 9.117894034638208548514808203742, 9.256691510852288615675806117776, 9.886707849578587066687733658491
