| L(s) = 1 | − 2-s + 3.30·3-s + 4-s + 5-s − 3.30·6-s − 0.302·7-s − 8-s + 7.90·9-s − 10-s − 5.30·11-s + 3.30·12-s − 0.302·13-s + 0.302·14-s + 3.30·15-s + 16-s − 3.90·17-s − 7.90·18-s − 4.90·19-s + 20-s − 1.00·21-s + 5.30·22-s − 23-s − 3.30·24-s + 25-s + 0.302·26-s + 16.2·27-s − 0.302·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.90·3-s + 0.5·4-s + 0.447·5-s − 1.34·6-s − 0.114·7-s − 0.353·8-s + 2.63·9-s − 0.316·10-s − 1.59·11-s + 0.953·12-s − 0.0839·13-s + 0.0809·14-s + 0.852·15-s + 0.250·16-s − 0.947·17-s − 1.86·18-s − 1.12·19-s + 0.223·20-s − 0.218·21-s + 1.13·22-s − 0.208·23-s − 0.674·24-s + 0.200·25-s + 0.0593·26-s + 3.11·27-s − 0.0572·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593170128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593170128\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 7 | \( 1 + 0.302T + 7T^{2} \) |
| 11 | \( 1 + 5.30T + 11T^{2} \) |
| 13 | \( 1 + 0.302T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 2.90T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 - 5.21T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 - 3.21T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 6.51T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2.69T + 97T^{2} \) |
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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
−12.54562329491617748097097436723, −10.71046230091007479456466608837, −10.06568592678611651751133565890, −9.132013969731634669302277945894, −8.341857495284555451505842873076, −7.67615958829551750295562810625, −6.49361484218236037802429356517, −4.55549946053216157588369309046, −2.91208024671020648339608776976, −2.11808372085269199478702255532,
2.11808372085269199478702255532, 2.91208024671020648339608776976, 4.55549946053216157588369309046, 6.49361484218236037802429356517, 7.67615958829551750295562810625, 8.341857495284555451505842873076, 9.132013969731634669302277945894, 10.06568592678611651751133565890, 10.71046230091007479456466608837, 12.54562329491617748097097436723
