L(s) = 1 | − 2-s + 1.45·3-s + 4-s + 5-s − 1.45·6-s + 3.96·7-s − 8-s − 0.892·9-s − 10-s − 3.61·11-s + 1.45·12-s − 3.92·13-s − 3.96·14-s + 1.45·15-s + 16-s + 4.03·17-s + 0.892·18-s + 1.77·19-s + 20-s + 5.75·21-s + 3.61·22-s + 6.25·23-s − 1.45·24-s + 25-s + 3.92·26-s − 5.65·27-s + 3.96·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.838·3-s + 0.5·4-s + 0.447·5-s − 0.592·6-s + 1.49·7-s − 0.353·8-s − 0.297·9-s − 0.316·10-s − 1.09·11-s + 0.419·12-s − 1.08·13-s − 1.05·14-s + 0.374·15-s + 0.250·16-s + 0.979·17-s + 0.210·18-s + 0.406·19-s + 0.223·20-s + 1.25·21-s + 0.770·22-s + 1.30·23-s − 0.296·24-s + 0.200·25-s + 0.770·26-s − 1.08·27-s + 0.749·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.327405548\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.327405548\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 - 1.45T + 3T^{2} \) |
| 7 | \( 1 - 3.96T + 7T^{2} \) |
| 11 | \( 1 + 3.61T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 - 4.03T + 17T^{2} \) |
| 19 | \( 1 - 1.77T + 19T^{2} \) |
| 23 | \( 1 - 6.25T + 23T^{2} \) |
| 29 | \( 1 - 0.523T + 29T^{2} \) |
| 31 | \( 1 + 2.59T + 31T^{2} \) |
| 37 | \( 1 - 3.94T + 37T^{2} \) |
| 41 | \( 1 + 2.98T + 41T^{2} \) |
| 43 | \( 1 + 0.307T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 4.81T + 59T^{2} \) |
| 61 | \( 1 + 4.91T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 5.27T + 71T^{2} \) |
| 73 | \( 1 + 6.21T + 73T^{2} \) |
| 79 | \( 1 - 5.85T + 79T^{2} \) |
| 83 | \( 1 + 1.10T + 83T^{2} \) |
| 89 | \( 1 + 9.25T + 89T^{2} \) |
| 97 | \( 1 - 1.27T + 97T^{2} \) |
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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
−8.162913722246496411391559470656, −7.44401559463960371019470498548, −7.24162882045254723204897970380, −5.72863734012826772380856234873, −5.33180434692991407299139750961, −4.57834986287218262136013959725, −3.27739864952021130535656055997, −2.55553528310786255717070533868, −1.98032817529319181040578601663, −0.862316330659185684695759097095,
0.862316330659185684695759097095, 1.98032817529319181040578601663, 2.55553528310786255717070533868, 3.27739864952021130535656055997, 4.57834986287218262136013959725, 5.33180434692991407299139750961, 5.72863734012826772380856234873, 7.24162882045254723204897970380, 7.44401559463960371019470498548, 8.162913722246496411391559470656