| L(s) = 1 | + 2-s + 3.15·3-s + 4-s − 5-s + 3.15·6-s − 0.586·7-s + 8-s + 6.93·9-s − 10-s + 1.70·11-s + 3.15·12-s + 6.24·13-s − 0.586·14-s − 3.15·15-s + 16-s + 2.23·17-s + 6.93·18-s − 1.53·19-s − 20-s − 1.84·21-s + 1.70·22-s + 3.15·24-s + 25-s + 6.24·26-s + 12.3·27-s − 0.586·28-s − 0.273·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.81·3-s + 0.5·4-s − 0.447·5-s + 1.28·6-s − 0.221·7-s + 0.353·8-s + 2.31·9-s − 0.316·10-s + 0.513·11-s + 0.909·12-s + 1.73·13-s − 0.156·14-s − 0.813·15-s + 0.250·16-s + 0.541·17-s + 1.63·18-s − 0.352·19-s − 0.223·20-s − 0.403·21-s + 0.362·22-s + 0.643·24-s + 0.200·25-s + 1.22·26-s + 2.38·27-s − 0.110·28-s − 0.0508·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.537525277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.537525277\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 - 3.15T + 3T^{2} \) |
| 7 | \( 1 + 0.586T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 29 | \( 1 + 0.273T + 29T^{2} \) |
| 31 | \( 1 + 8.90T + 31T^{2} \) |
| 37 | \( 1 + 8.46T + 37T^{2} \) |
| 41 | \( 1 + 3.08T + 41T^{2} \) |
| 43 | \( 1 - 7.42T + 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 6.50T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 7.94T + 71T^{2} \) |
| 73 | \( 1 - 1.43T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 4.34T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 3.88T + 97T^{2} \) |
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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
−8.231665599837243126616247144694, −7.54032471275707740019774943405, −6.87840869778376847345209344099, −6.14908405416468628866530281666, −5.11623118199265016039457383961, −3.99716917500655441775449869458, −3.68238878946379513873601438882, −3.17580021330184224954817697362, −2.08241681009634084066801733708, −1.30170981119584858069118064641,
1.30170981119584858069118064641, 2.08241681009634084066801733708, 3.17580021330184224954817697362, 3.68238878946379513873601438882, 3.99716917500655441775449869458, 5.11623118199265016039457383961, 6.14908405416468628866530281666, 6.87840869778376847345209344099, 7.54032471275707740019774943405, 8.231665599837243126616247144694
