| L(s) = 1 | + 1.61·2-s + 2.61·3-s + 0.618·4-s − 1.23·5-s + 4.23·6-s − 0.618·7-s − 2.23·8-s + 3.85·9-s − 2.00·10-s − 3·11-s + 1.61·12-s + 1.76·13-s − 1.00·14-s − 3.23·15-s − 4.85·16-s + 4.38·17-s + 6.23·18-s − 5.85·19-s − 0.763·20-s − 1.61·21-s − 4.85·22-s + 6.23·23-s − 5.85·24-s − 3.47·25-s + 2.85·26-s + 2.23·27-s − 0.381·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 1.51·3-s + 0.309·4-s − 0.552·5-s + 1.72·6-s − 0.233·7-s − 0.790·8-s + 1.28·9-s − 0.632·10-s − 0.904·11-s + 0.467·12-s + 0.489·13-s − 0.267·14-s − 0.835·15-s − 1.21·16-s + 1.06·17-s + 1.46·18-s − 1.34·19-s − 0.170·20-s − 0.353·21-s − 1.03·22-s + 1.30·23-s − 1.19·24-s − 0.694·25-s + 0.559·26-s + 0.430·27-s − 0.0721·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.541629624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.541629624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 211 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 - 0.0901T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 + 0.618T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 - 6.70T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 9.09T + 73T^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 - 4.14T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.94688230149341132220237572836, −11.73793555735758583432511131487, −10.40572907488464236404775667184, −9.179824028246581985165728588630, −8.366610100307217069606563423091, −7.43849445532593941380600467855, −5.95565940507301218652034508117, −4.53165472346215825844321097401, −3.53059258412759958144996583971, −2.65431700043165294476165593416,
2.65431700043165294476165593416, 3.53059258412759958144996583971, 4.53165472346215825844321097401, 5.95565940507301218652034508117, 7.43849445532593941380600467855, 8.366610100307217069606563423091, 9.179824028246581985165728588630, 10.40572907488464236404775667184, 11.73793555735758583432511131487, 12.94688230149341132220237572836
