| L(s) = 1 | + 1.61·2-s + 0.593·4-s + 0.989·5-s − 7-s − 2.26·8-s + 1.59·10-s + 3.22·11-s + 5.83·13-s − 1.61·14-s − 4.83·16-s + 0.989·17-s + 19-s + 0.586·20-s + 5.18·22-s + 6.50·23-s − 4.02·25-s + 9.39·26-s − 0.593·28-s + 2.23·29-s + 8.64·31-s − 3.25·32-s + 1.59·34-s − 0.989·35-s + 1.18·37-s + 1.61·38-s − 2.24·40-s − 3.22·41-s + ⋯ |
| L(s) = 1 | + 1.13·2-s + 0.296·4-s + 0.442·5-s − 0.377·7-s − 0.800·8-s + 0.503·10-s + 0.971·11-s + 1.61·13-s − 0.430·14-s − 1.20·16-s + 0.239·17-s + 0.229·19-s + 0.131·20-s + 1.10·22-s + 1.35·23-s − 0.804·25-s + 1.84·26-s − 0.112·28-s + 0.414·29-s + 1.55·31-s − 0.575·32-s + 0.273·34-s − 0.167·35-s + 0.195·37-s + 0.261·38-s − 0.354·40-s − 0.502·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.075714784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.075714784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 - 0.989T + 5T^{2} \) |
| 11 | \( 1 - 3.22T + 11T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 17 | \( 1 - 0.989T + 17T^{2} \) |
| 23 | \( 1 - 6.50T + 23T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 - 8.64T + 31T^{2} \) |
| 37 | \( 1 - 1.18T + 37T^{2} \) |
| 41 | \( 1 + 3.22T + 41T^{2} \) |
| 43 | \( 1 - 0.165T + 43T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 - 8.00T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 1.18T + 61T^{2} \) |
| 67 | \( 1 - 5.62T + 67T^{2} \) |
| 71 | \( 1 + 6.76T + 71T^{2} \) |
| 73 | \( 1 + 8.85T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 8.67T + 83T^{2} \) |
| 89 | \( 1 + 3.22T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.645225300560866388818253624241, −9.014406702074141143368610336390, −8.235118579724050059972506326814, −6.78010027530487281159536491343, −6.26589371374337675931480052692, −5.54580355305659230935800502068, −4.50265286160104569276191232781, −3.67007988110464106903088404985, −2.88533573746528464865166161158, −1.24849247582316168855581211003,
1.24849247582316168855581211003, 2.88533573746528464865166161158, 3.67007988110464106903088404985, 4.50265286160104569276191232781, 5.54580355305659230935800502068, 6.26589371374337675931480052692, 6.78010027530487281159536491343, 8.235118579724050059972506326814, 9.014406702074141143368610336390, 9.645225300560866388818253624241
