| L(s) = 1 | − 2·7-s − 3·9-s + 4·13-s + 16·19-s − 6·25-s + 16·31-s − 4·37-s − 8·43-s + 3·49-s − 12·61-s + 6·63-s − 8·67-s + 20·73-s + 32·79-s + 9·81-s − 8·91-s − 12·97-s − 32·103-s − 20·109-s − 12·117-s − 6·121-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 9-s + 1.10·13-s + 3.67·19-s − 6/5·25-s + 2.87·31-s − 0.657·37-s − 1.21·43-s + 3/7·49-s − 1.53·61-s + 0.755·63-s − 0.977·67-s + 2.34·73-s + 3.60·79-s + 81-s − 0.838·91-s − 1.21·97-s − 3.15·103-s − 1.91·109-s − 1.10·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.159404707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.159404707\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−10.60224244605800499935670328540, −9.886707849578587066687733658491, −9.457349313423188294666539437989, −9.256691510852288615675806117776, −8.188420417684287628984093773695, −8.093537689649372229665039418958, −7.40397429472305265033368100557, −6.42262084813563967264199691637, −6.35373904523042773985731435784, −5.29340219049011126125230323142, −5.22316409435338364257345412913, −3.89550776474805329328185747620, −3.26735667825841356907489548421, −2.79183800612725741835263061431, −1.15037892508990574535308118020,
1.15037892508990574535308118020, 2.79183800612725741835263061431, 3.26735667825841356907489548421, 3.89550776474805329328185747620, 5.22316409435338364257345412913, 5.29340219049011126125230323142, 6.35373904523042773985731435784, 6.42262084813563967264199691637, 7.40397429472305265033368100557, 8.093537689649372229665039418958, 8.188420417684287628984093773695, 9.256691510852288615675806117776, 9.457349313423188294666539437989, 9.886707849578587066687733658491, 10.60224244605800499935670328540
