| L(s) = 1 | + 1.81·2-s + 1.28·4-s + 2.10·5-s + 7-s − 1.28·8-s + 3.81·10-s + 0.186·13-s + 1.81·14-s − 4.91·16-s + 2.10·17-s + 4.52·19-s + 2.71·20-s + 1.94·23-s − 0.578·25-s + 0.338·26-s + 1.28·28-s − 0.186·29-s + 4.20·31-s − 6.33·32-s + 3.81·34-s + 2.10·35-s − 1.28·37-s + 8.20·38-s − 2.71·40-s − 8.91·41-s + 10.9·43-s + 3.52·46-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.644·4-s + 0.940·5-s + 0.377·7-s − 0.455·8-s + 1.20·10-s + 0.0516·13-s + 0.484·14-s − 1.22·16-s + 0.509·17-s + 1.03·19-s + 0.606·20-s + 0.405·23-s − 0.115·25-s + 0.0662·26-s + 0.243·28-s − 0.0346·29-s + 0.755·31-s − 1.12·32-s + 0.654·34-s + 0.355·35-s − 0.211·37-s + 1.33·38-s − 0.428·40-s − 1.39·41-s + 1.67·43-s + 0.520·46-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
| \(L(1)\) |
\(\approx\) |
\(5.139616984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.139616984\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.81T + 2T^{2} \) |
| 5 | \( 1 - 2.10T + 5T^{2} \) |
| 13 | \( 1 - 0.186T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 - 4.52T + 19T^{2} \) |
| 23 | \( 1 - 1.94T + 23T^{2} \) |
| 29 | \( 1 + 0.186T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 + 1.28T + 37T^{2} \) |
| 41 | \( 1 + 8.91T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 9.01T + 47T^{2} \) |
| 53 | \( 1 - 6.71T + 53T^{2} \) |
| 59 | \( 1 - 4.44T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 5.07T + 67T^{2} \) |
| 71 | \( 1 + 6.72T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 7.45T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 0.946T + 89T^{2} \) |
| 97 | \( 1 - 7.27T + 97T^{2} \) |
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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
−7.63549807470390278461747294412, −7.02098577536126392120722147338, −6.14455579827531752801754494083, −5.63155403376443342310899142277, −5.17114512883715914405550758343, −4.40188471031630479817861266736, −3.61633066854771695604746068431, −2.83156736140648020759787145366, −2.08888940217803112141084603907, −0.965183611406877851471093011788,
0.965183611406877851471093011788, 2.08888940217803112141084603907, 2.83156736140648020759787145366, 3.61633066854771695604746068431, 4.40188471031630479817861266736, 5.17114512883715914405550758343, 5.63155403376443342310899142277, 6.14455579827531752801754494083, 7.02098577536126392120722147338, 7.63549807470390278461747294412
