| L(s) = 1 | + 2-s − 2.14·3-s + 4-s − 1.60·5-s − 2.14·6-s − 2.89·7-s + 8-s + 1.60·9-s − 1.60·10-s − 2.14·12-s − 4.89·13-s − 2.89·14-s + 3.43·15-s + 16-s + 5.74·17-s + 1.60·18-s + 19-s − 1.60·20-s + 6.20·21-s − 7.74·23-s − 2.14·24-s − 2.43·25-s − 4.89·26-s + 3.00·27-s − 2.89·28-s − 5.34·29-s + 3.43·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.23·3-s + 0.5·4-s − 0.716·5-s − 0.875·6-s − 1.09·7-s + 0.353·8-s + 0.533·9-s − 0.506·10-s − 0.619·12-s − 1.35·13-s − 0.772·14-s + 0.886·15-s + 0.250·16-s + 1.39·17-s + 0.377·18-s + 0.229·19-s − 0.358·20-s + 1.35·21-s − 1.61·23-s − 0.437·24-s − 0.487·25-s − 0.959·26-s + 0.577·27-s − 0.546·28-s − 0.993·29-s + 0.627·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6006735829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6006735829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.14T + 3T^{2} \) |
| 5 | \( 1 + 1.60T + 5T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 23 | \( 1 + 7.74T + 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 + 7.43T + 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 7.08T + 47T^{2} \) |
| 53 | \( 1 - 8.32T + 53T^{2} \) |
| 59 | \( 1 + 2.54T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 3.74T + 73T^{2} \) |
| 79 | \( 1 - 8.69T + 79T^{2} \) |
| 83 | \( 1 - 6.10T + 83T^{2} \) |
| 89 | \( 1 - 3.20T + 89T^{2} \) |
| 97 | \( 1 + 6.98T + 97T^{2} \) |
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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
−7.87672109323145568215943172123, −7.47089459622548593629933736718, −6.74213818813617001930405869079, −5.83525763078345636404303867827, −5.59083887467222899919622946134, −4.69687699668499012400286236153, −3.79734070238686811712256055200, −3.22074542336254436137027349507, −2.00291871005991870335858583302, −0.38907638635255401009910198760,
0.38907638635255401009910198760, 2.00291871005991870335858583302, 3.22074542336254436137027349507, 3.79734070238686811712256055200, 4.69687699668499012400286236153, 5.59083887467222899919622946134, 5.83525763078345636404303867827, 6.74213818813617001930405869079, 7.47089459622548593629933736718, 7.87672109323145568215943172123
