L(s) = 1 | + 81·3-s − 1.31e3·5-s + 4.48e3·7-s + 6.56e3·9-s − 1.47e3·11-s − 1.51e5·13-s − 1.06e5·15-s + 1.08e5·17-s − 5.93e5·19-s + 3.62e5·21-s + 9.69e5·23-s − 2.26e5·25-s + 5.31e5·27-s − 6.64e6·29-s − 7.07e6·31-s − 1.19e5·33-s − 5.88e6·35-s − 7.47e6·37-s − 1.22e7·39-s − 4.35e6·41-s + 4.35e6·43-s − 8.62e6·45-s − 2.83e7·47-s − 2.02e7·49-s + 8.76e6·51-s + 1.61e7·53-s + 1.93e6·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.940·5-s + 0.705·7-s + 1/3·9-s − 0.0303·11-s − 1.47·13-s − 0.542·15-s + 0.314·17-s − 1.04·19-s + 0.407·21-s + 0.722·23-s − 0.115·25-s + 0.192·27-s − 1.74·29-s − 1.37·31-s − 0.0175·33-s − 0.663·35-s − 0.655·37-s − 0.849·39-s − 0.240·41-s + 0.194·43-s − 0.313·45-s − 0.846·47-s − 0.502·49-s + 0.181·51-s + 0.280·53-s + 0.0285·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{4} T \) |
good | 5 | \( 1 + 1314 T + p^{9} T^{2} \) |
| 7 | \( 1 - 640 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 1476 T + p^{9} T^{2} \) |
| 13 | \( 1 + 151522 T + p^{9} T^{2} \) |
| 17 | \( 1 - 108162 T + p^{9} T^{2} \) |
| 19 | \( 1 + 593084 T + p^{9} T^{2} \) |
| 23 | \( 1 - 969480 T + p^{9} T^{2} \) |
| 29 | \( 1 + 6642522 T + p^{9} T^{2} \) |
| 31 | \( 1 + 7070600 T + p^{9} T^{2} \) |
| 37 | \( 1 + 7472410 T + p^{9} T^{2} \) |
| 41 | \( 1 + 4350150 T + p^{9} T^{2} \) |
| 43 | \( 1 - 4358716 T + p^{9} T^{2} \) |
| 47 | \( 1 + 28309248 T + p^{9} T^{2} \) |
| 53 | \( 1 - 16111710 T + p^{9} T^{2} \) |
| 59 | \( 1 - 86075964 T + p^{9} T^{2} \) |
| 61 | \( 1 - 32213918 T + p^{9} T^{2} \) |
| 67 | \( 1 + 99531452 T + p^{9} T^{2} \) |
| 71 | \( 1 - 44170488 T + p^{9} T^{2} \) |
| 73 | \( 1 + 23560630 T + p^{9} T^{2} \) |
| 79 | \( 1 - 401754760 T + p^{9} T^{2} \) |
| 83 | \( 1 - 744528708 T + p^{9} T^{2} \) |
| 89 | \( 1 - 769871034 T + p^{9} T^{2} \) |
| 97 | \( 1 - 907130882 T + p^{9} T^{2} \) |
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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
−13.06267565558064360819841292317, −11.97216163855752854798208673036, −10.82845430029998821706160804127, −9.335605634997783277221241487177, −8.042785218603163251396388606649, −7.20541553487380733467907169907, −5.03647276094300303888600772149, −3.70994467330358667034517268199, −2.03244412601164472223955114225, 0,
2.03244412601164472223955114225, 3.70994467330358667034517268199, 5.03647276094300303888600772149, 7.20541553487380733467907169907, 8.042785218603163251396388606649, 9.335605634997783277221241487177, 10.82845430029998821706160804127, 11.97216163855752854798208673036, 13.06267565558064360819841292317