| L(s) = 1 | + 2-s − 0.687·3-s + 4-s + 5-s − 0.687·6-s − 3.00·7-s + 8-s − 2.52·9-s + 10-s − 3.24·11-s − 0.687·12-s + 4.42·13-s − 3.00·14-s − 0.687·15-s + 16-s − 7.05·17-s − 2.52·18-s − 3.61·19-s + 20-s + 2.06·21-s − 3.24·22-s − 0.687·24-s + 25-s + 4.42·26-s + 3.80·27-s − 3.00·28-s − 3.99·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.397·3-s + 0.5·4-s + 0.447·5-s − 0.280·6-s − 1.13·7-s + 0.353·8-s − 0.842·9-s + 0.316·10-s − 0.977·11-s − 0.198·12-s + 1.22·13-s − 0.803·14-s − 0.177·15-s + 0.250·16-s − 1.71·17-s − 0.595·18-s − 0.829·19-s + 0.223·20-s + 0.451·21-s − 0.690·22-s − 0.140·24-s + 0.200·25-s + 0.867·26-s + 0.731·27-s − 0.568·28-s − 0.741·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.845592071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.845592071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.687T + 3T^{2} \) |
| 7 | \( 1 + 3.00T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 13 | \( 1 - 4.42T + 13T^{2} \) |
| 17 | \( 1 + 7.05T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 29 | \( 1 + 3.99T + 29T^{2} \) |
| 31 | \( 1 - 1.99T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 6.41T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 7.64T + 59T^{2} \) |
| 61 | \( 1 - 7.00T + 61T^{2} \) |
| 67 | \( 1 + 3.21T + 67T^{2} \) |
| 71 | \( 1 - 8.14T + 71T^{2} \) |
| 73 | \( 1 - 6.95T + 73T^{2} \) |
| 79 | \( 1 + 4.39T + 79T^{2} \) |
| 83 | \( 1 + 4.45T + 83T^{2} \) |
| 89 | \( 1 - 5.09T + 89T^{2} \) |
| 97 | \( 1 + 4.03T + 97T^{2} \) |
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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
−8.217181846056013355345137365417, −7.23111052189533762450318685166, −6.38624986839357382503844087225, −6.04418657071165137815754489225, −5.52925075472608631642544341137, −4.45823962562920308285489316169, −3.82124916544509246753993364317, −2.69180545421028201132620742619, −2.34793693354499784451364223505, −0.63447158576614530560844739963,
0.63447158576614530560844739963, 2.34793693354499784451364223505, 2.69180545421028201132620742619, 3.82124916544509246753993364317, 4.45823962562920308285489316169, 5.52925075472608631642544341137, 6.04418657071165137815754489225, 6.38624986839357382503844087225, 7.23111052189533762450318685166, 8.217181846056013355345137365417
