| L(s) = 1 | + 3-s + 2·5-s + 9-s + 6·13-s + 2·15-s + 6·17-s + 8·19-s − 25-s + 27-s + 6·29-s + 6·37-s + 6·39-s − 10·41-s − 8·43-s + 2·45-s + 6·51-s + 6·53-s + 8·57-s + 4·59-s − 2·61-s + 12·65-s + 12·67-s + 8·71-s + 2·73-s − 75-s − 4·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.66·13-s + 0.516·15-s + 1.45·17-s + 1.83·19-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.986·37-s + 0.960·39-s − 1.56·41-s − 1.21·43-s + 0.298·45-s + 0.840·51-s + 0.824·53-s + 1.05·57-s + 0.520·59-s − 0.256·61-s + 1.48·65-s + 1.46·67-s + 0.949·71-s + 0.234·73-s − 0.115·75-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 284592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 284592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.604000744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.604000744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.92680889010247, −12.13506726454039, −11.94031847804282, −11.38625335186084, −10.86832946966851, −10.22283615066680, −9.878003389920687, −9.682832493260709, −9.047611649462292, −8.555123422200858, −8.041632479041275, −7.851827192556627, −6.995278293456789, −6.707444022679189, −5.990551276024550, −5.706590998892773, −5.168463928708999, −4.697282392007429, −3.782270319854920, −3.410083493914127, −3.143833700104064, −2.326498817462529, −1.746639538549003, −1.091127296329411, −0.8383977030035573,
0.8383977030035573, 1.091127296329411, 1.746639538549003, 2.326498817462529, 3.143833700104064, 3.410083493914127, 3.782270319854920, 4.697282392007429, 5.168463928708999, 5.706590998892773, 5.990551276024550, 6.707444022679189, 6.995278293456789, 7.851827192556627, 8.041632479041275, 8.555123422200858, 9.047611649462292, 9.682832493260709, 9.878003389920687, 10.22283615066680, 10.86832946966851, 11.38625335186084, 11.94031847804282, 12.13506726454039, 12.92680889010247
