| L(s) = 1 | − 5-s − 11-s + 6·13-s − 4·17-s − 2·19-s + 4·23-s + 25-s + 6·29-s − 2·37-s + 8·41-s − 4·43-s + 14·53-s + 55-s + 6·59-s + 10·61-s − 6·65-s + 4·67-s + 4·71-s − 8·73-s − 4·79-s + 6·83-s + 4·85-s − 8·89-s + 2·95-s + 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 1.66·13-s − 0.970·17-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.328·37-s + 1.24·41-s − 0.609·43-s + 1.92·53-s + 0.134·55-s + 0.781·59-s + 1.28·61-s − 0.744·65-s + 0.488·67-s + 0.474·71-s − 0.936·73-s − 0.450·79-s + 0.658·83-s + 0.433·85-s − 0.847·89-s + 0.205·95-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.881229570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.881229570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.48761265761168, −11.96386933259087, −11.40900649978236, −11.18232067431748, −10.71717107824150, −10.31359825704940, −9.845093530449973, −9.070371380658296, −8.761633564846056, −8.462325761268240, −8.038008763731720, −7.380513270022510, −6.847725881092572, −6.569715828236331, −6.023029006507355, −5.467253743253446, −4.993417187864808, −4.342430339703762, −3.961845016489580, −3.547143083489177, −2.793029657345824, −2.431887393342528, −1.663136201418979, −0.9720689704155269, −0.5163136948507769,
0.5163136948507769, 0.9720689704155269, 1.663136201418979, 2.431887393342528, 2.793029657345824, 3.547143083489177, 3.961845016489580, 4.342430339703762, 4.993417187864808, 5.467253743253446, 6.023029006507355, 6.569715828236331, 6.847725881092572, 7.380513270022510, 8.038008763731720, 8.462325761268240, 8.761633564846056, 9.070371380658296, 9.845093530449973, 10.31359825704940, 10.71717107824150, 11.18232067431748, 11.40900649978236, 11.96386933259087, 12.48761265761168
