L(s) = 1 | − 2.23·2-s − 3-s + 3.00·4-s − 0.618·5-s + 2.23·6-s − 7-s − 2.23·8-s + 9-s + 1.38·10-s − 3.00·12-s − 3.23·13-s + 2.23·14-s + 0.618·15-s − 0.999·16-s − 4.85·17-s − 2.23·18-s − 2.85·19-s − 1.85·20-s + 21-s + 4.38·23-s + 2.23·24-s − 4.61·25-s + 7.23·26-s − 27-s − 3.00·28-s + 6·29-s − 1.38·30-s + ⋯ |
L(s) = 1 | − 1.58·2-s − 0.577·3-s + 1.50·4-s − 0.276·5-s + 0.912·6-s − 0.377·7-s − 0.790·8-s + 0.333·9-s + 0.437·10-s − 0.866·12-s − 0.897·13-s + 0.597·14-s + 0.159·15-s − 0.249·16-s − 1.17·17-s − 0.527·18-s − 0.654·19-s − 0.414·20-s + 0.218·21-s + 0.913·23-s + 0.456·24-s − 0.923·25-s + 1.41·26-s − 0.192·27-s − 0.566·28-s + 1.11·29-s − 0.252·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2849916490\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2849916490\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 + 2.85T + 19T^{2} \) |
| 23 | \( 1 - 4.38T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 - 4.61T + 37T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 + 9.70T + 43T^{2} \) |
| 47 | \( 1 - 4.47T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 - 3.23T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 5.61T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.848433022841399156531359095853, −8.355754819927098073687663906451, −7.34809756143244920901064055098, −6.90895612382047214695359970370, −6.18082924958431772662663023560, −4.99845124870622948749510710670, −4.18039010281958255531008494955, −2.76179934789303761200821768323, −1.78910380924120353574255154314, −0.42898484245146638831882434050,
0.42898484245146638831882434050, 1.78910380924120353574255154314, 2.76179934789303761200821768323, 4.18039010281958255531008494955, 4.99845124870622948749510710670, 6.18082924958431772662663023560, 6.90895612382047214695359970370, 7.34809756143244920901064055098, 8.355754819927098073687663906451, 8.848433022841399156531359095853