| L(s) = 1 | − 2-s − 3-s + 4-s − 2.02·5-s + 6-s − 1.70·7-s − 8-s + 9-s + 2.02·10-s − 0.223·11-s − 12-s + 4.73·13-s + 1.70·14-s + 2.02·15-s + 16-s − 2.06·17-s − 18-s − 19-s − 2.02·20-s + 1.70·21-s + 0.223·22-s + 4.59·23-s + 24-s − 0.883·25-s − 4.73·26-s − 27-s − 1.70·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.907·5-s + 0.408·6-s − 0.642·7-s − 0.353·8-s + 0.333·9-s + 0.641·10-s − 0.0673·11-s − 0.288·12-s + 1.31·13-s + 0.454·14-s + 0.523·15-s + 0.250·16-s − 0.500·17-s − 0.235·18-s − 0.229·19-s − 0.453·20-s + 0.371·21-s + 0.0476·22-s + 0.957·23-s + 0.204·24-s − 0.176·25-s − 0.929·26-s − 0.192·27-s − 0.321·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6434025546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6434025546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 2.02T + 5T^{2} \) |
| 7 | \( 1 + 1.70T + 7T^{2} \) |
| 11 | \( 1 + 0.223T + 11T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 + 2.06T + 17T^{2} \) |
| 23 | \( 1 - 4.59T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 + 6.41T + 31T^{2} \) |
| 37 | \( 1 - 5.05T + 37T^{2} \) |
| 41 | \( 1 + 5.36T + 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 59 | \( 1 - 9.34T + 59T^{2} \) |
| 61 | \( 1 + 9.93T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 7.18T + 71T^{2} \) |
| 73 | \( 1 + 7.93T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 3.88T + 83T^{2} \) |
| 89 | \( 1 + 2.21T + 89T^{2} \) |
| 97 | \( 1 + 3.71T + 97T^{2} \) |
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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
−8.151953293316770578317671726519, −7.35982215352783307463961696361, −6.66797905109862439895233279515, −6.22338966833705210668297596213, −5.31274039419000401431167298191, −4.34395079554166944880088064946, −3.61751117141891910614308658539, −2.82943639708352003739474483681, −1.51576833862843940560399500814, −0.50269612982984683248568409238,
0.50269612982984683248568409238, 1.51576833862843940560399500814, 2.82943639708352003739474483681, 3.61751117141891910614308658539, 4.34395079554166944880088064946, 5.31274039419000401431167298191, 6.22338966833705210668297596213, 6.66797905109862439895233279515, 7.35982215352783307463961696361, 8.151953293316770578317671726519
