| L(s) = 1 | − 0.574·2-s − 3-s − 1.66·4-s − 2.72·5-s + 0.574·6-s + 0.829·7-s + 2.10·8-s + 9-s + 1.56·10-s + 4.58·11-s + 1.66·12-s − 13-s − 0.476·14-s + 2.72·15-s + 2.12·16-s − 2.51·17-s − 0.574·18-s + 3.53·19-s + 4.55·20-s − 0.829·21-s − 2.63·22-s + 0.402·23-s − 2.10·24-s + 2.42·25-s + 0.574·26-s − 27-s − 1.38·28-s + ⋯ |
| L(s) = 1 | − 0.406·2-s − 0.577·3-s − 0.834·4-s − 1.21·5-s + 0.234·6-s + 0.313·7-s + 0.745·8-s + 0.333·9-s + 0.495·10-s + 1.38·11-s + 0.482·12-s − 0.277·13-s − 0.127·14-s + 0.703·15-s + 0.532·16-s − 0.608·17-s − 0.135·18-s + 0.810·19-s + 1.01·20-s − 0.181·21-s − 0.561·22-s + 0.0838·23-s − 0.430·24-s + 0.484·25-s + 0.112·26-s − 0.192·27-s − 0.261·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7150593346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7150593346\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 0.574T + 2T^{2} \) |
| 5 | \( 1 + 2.72T + 5T^{2} \) |
| 7 | \( 1 - 0.829T + 7T^{2} \) |
| 11 | \( 1 - 4.58T + 11T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 19 | \( 1 - 3.53T + 19T^{2} \) |
| 23 | \( 1 - 0.402T + 23T^{2} \) |
| 29 | \( 1 - 6.35T + 29T^{2} \) |
| 31 | \( 1 + 2.35T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 - 0.900T + 41T^{2} \) |
| 43 | \( 1 - 5.57T + 43T^{2} \) |
| 47 | \( 1 + 5.90T + 47T^{2} \) |
| 53 | \( 1 - 5.35T + 53T^{2} \) |
| 59 | \( 1 + 2.99T + 59T^{2} \) |
| 61 | \( 1 - 0.725T + 61T^{2} \) |
| 67 | \( 1 + 5.01T + 67T^{2} \) |
| 71 | \( 1 + 1.44T + 71T^{2} \) |
| 73 | \( 1 + 7.03T + 73T^{2} \) |
| 79 | \( 1 + 8.55T + 79T^{2} \) |
| 83 | \( 1 - 1.90T + 83T^{2} \) |
| 89 | \( 1 + 0.222T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.475459131091283937710307390365, −7.74534171028917823765058870019, −7.15464554154615527349833866535, −6.34455220939587283764340275648, −5.32427083581104826582359392988, −4.46031315942006905416706661659, −4.14951985539929055624567134642, −3.19548216553437337839894711415, −1.51616882939453407818418663150, −0.57903319632653836530823137688,
0.57903319632653836530823137688, 1.51616882939453407818418663150, 3.19548216553437337839894711415, 4.14951985539929055624567134642, 4.46031315942006905416706661659, 5.32427083581104826582359392988, 6.34455220939587283764340275648, 7.15464554154615527349833866535, 7.74534171028917823765058870019, 8.475459131091283937710307390365
