| L(s) = 1 | − 2-s − 3.33·3-s + 4-s − 3.05·5-s + 3.33·6-s + 7-s − 8-s + 8.10·9-s + 3.05·10-s − 3.33·12-s − 2.01·13-s − 14-s + 10.1·15-s + 16-s − 3.87·17-s − 8.10·18-s + 0.322·19-s − 3.05·20-s − 3.33·21-s + 0.774·23-s + 3.33·24-s + 4.36·25-s + 2.01·26-s − 17.0·27-s + 28-s + 0.302·29-s − 10.1·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.92·3-s + 0.5·4-s − 1.36·5-s + 1.36·6-s + 0.377·7-s − 0.353·8-s + 2.70·9-s + 0.967·10-s − 0.962·12-s − 0.557·13-s − 0.267·14-s + 2.63·15-s + 0.250·16-s − 0.938·17-s − 1.91·18-s + 0.0739·19-s − 0.684·20-s − 0.727·21-s + 0.161·23-s + 0.680·24-s + 0.872·25-s + 0.394·26-s − 3.27·27-s + 0.188·28-s + 0.0561·29-s − 1.86·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2075660629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2075660629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 3.33T + 3T^{2} \) |
| 5 | \( 1 + 3.05T + 5T^{2} \) |
| 13 | \( 1 + 2.01T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 - 0.322T + 19T^{2} \) |
| 23 | \( 1 - 0.774T + 23T^{2} \) |
| 29 | \( 1 - 0.302T + 29T^{2} \) |
| 31 | \( 1 + 4.77T + 31T^{2} \) |
| 37 | \( 1 + 5.37T + 37T^{2} \) |
| 41 | \( 1 + 6.10T + 41T^{2} \) |
| 43 | \( 1 + 8.28T + 43T^{2} \) |
| 47 | \( 1 - 1.21T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 2.22T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 5.53T + 73T^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 + 0.308T + 83T^{2} \) |
| 89 | \( 1 - 4.69T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−9.477895317406816673066650384877, −8.449462255902523974702683802792, −7.53581353354214975502667717883, −7.03043945329817845037426796404, −6.30656181257008771126620918623, −5.18720859092123080575178376554, −4.61674270908285028940023965830, −3.61293644289654804248651054659, −1.74973174460577097972633912513, −0.38350272159246642343175377113,
0.38350272159246642343175377113, 1.74973174460577097972633912513, 3.61293644289654804248651054659, 4.61674270908285028940023965830, 5.18720859092123080575178376554, 6.30656181257008771126620918623, 7.03043945329817845037426796404, 7.53581353354214975502667717883, 8.449462255902523974702683802792, 9.477895317406816673066650384877
