| L(s) = 1 | − 0.777·2-s − 1.39·4-s + 0.222·5-s + 7-s + 2.63·8-s − 0.173·10-s − 6.52·13-s − 0.777·14-s + 0.738·16-s − 4.33·17-s − 2.91·19-s − 0.310·20-s + 3.89·23-s − 4.95·25-s + 5.07·26-s − 1.39·28-s − 3.77·29-s − 6.88·31-s − 5.85·32-s + 3.37·34-s + 0.222·35-s − 5.65·37-s + 2.26·38-s + 0.587·40-s + 1.33·41-s + 4.70·43-s − 3.03·46-s + ⋯ |
| L(s) = 1 | − 0.549·2-s − 0.697·4-s + 0.0995·5-s + 0.377·7-s + 0.933·8-s − 0.0547·10-s − 1.81·13-s − 0.207·14-s + 0.184·16-s − 1.05·17-s − 0.668·19-s − 0.0694·20-s + 0.812·23-s − 0.990·25-s + 0.995·26-s − 0.263·28-s − 0.700·29-s − 1.23·31-s − 1.03·32-s + 0.577·34-s + 0.0376·35-s − 0.929·37-s + 0.367·38-s + 0.0928·40-s + 0.208·41-s + 0.717·43-s − 0.446·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5808066191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5808066191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.777T + 2T^{2} \) |
| 5 | \( 1 - 0.222T + 5T^{2} \) |
| 13 | \( 1 + 6.52T + 13T^{2} \) |
| 17 | \( 1 + 4.33T + 17T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 + 3.77T + 29T^{2} \) |
| 31 | \( 1 + 6.88T + 31T^{2} \) |
| 37 | \( 1 + 5.65T + 37T^{2} \) |
| 41 | \( 1 - 1.33T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 + 6.04T + 47T^{2} \) |
| 53 | \( 1 - 1.71T + 53T^{2} \) |
| 59 | \( 1 - 9.53T + 59T^{2} \) |
| 61 | \( 1 - 9.62T + 61T^{2} \) |
| 67 | \( 1 - 1.27T + 67T^{2} \) |
| 71 | \( 1 - 9.30T + 71T^{2} \) |
| 73 | \( 1 + 5.58T + 73T^{2} \) |
| 79 | \( 1 - 4.52T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 7.92T + 89T^{2} \) |
| 97 | \( 1 + 9.05T + 97T^{2} \) |
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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
−7.943033673577008168775129108725, −7.26140976801744730677174631632, −6.79024420056603834964535010443, −5.54550132698682753606864793233, −5.08793363740680971380090778588, −4.37218214300969828365020130897, −3.69615846170419875331401073204, −2.39452752985246507291431743720, −1.80836306475757154826217348687, −0.40450543110189062076705325119,
0.40450543110189062076705325119, 1.80836306475757154826217348687, 2.39452752985246507291431743720, 3.69615846170419875331401073204, 4.37218214300969828365020130897, 5.08793363740680971380090778588, 5.54550132698682753606864793233, 6.79024420056603834964535010443, 7.26140976801744730677174631632, 7.943033673577008168775129108725
