| L(s) = 1 | + 1.86·2-s + 3-s + 1.49·4-s − 3.24·5-s + 1.86·6-s − 2.84·7-s − 0.952·8-s + 9-s − 6.06·10-s + 5.52·11-s + 1.49·12-s + 6.06·13-s − 5.32·14-s − 3.24·15-s − 4.75·16-s − 17-s + 1.86·18-s − 5.16·19-s − 4.83·20-s − 2.84·21-s + 10.3·22-s − 2.22·23-s − 0.952·24-s + 5.54·25-s + 11.3·26-s + 27-s − 4.24·28-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.577·3-s + 0.745·4-s − 1.45·5-s + 0.762·6-s − 1.07·7-s − 0.336·8-s + 0.333·9-s − 1.91·10-s + 1.66·11-s + 0.430·12-s + 1.68·13-s − 1.42·14-s − 0.838·15-s − 1.18·16-s − 0.242·17-s + 0.440·18-s − 1.18·19-s − 1.08·20-s − 0.621·21-s + 2.20·22-s − 0.463·23-s − 0.194·24-s + 1.10·25-s + 2.22·26-s + 0.192·27-s − 0.802·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 1.86T + 2T^{2} \) |
| 5 | \( 1 + 3.24T + 5T^{2} \) |
| 7 | \( 1 + 2.84T + 7T^{2} \) |
| 11 | \( 1 - 5.52T + 11T^{2} \) |
| 13 | \( 1 - 6.06T + 13T^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 + 7.35T + 37T^{2} \) |
| 41 | \( 1 + 5.31T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 - 3.03T + 47T^{2} \) |
| 53 | \( 1 + 9.50T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 - 5.74T + 61T^{2} \) |
| 67 | \( 1 + 0.453T + 67T^{2} \) |
| 71 | \( 1 - 2.17T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 5.95T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.245197862632869505068991524883, −6.90792465486007281618118516960, −6.61735010542873618183438048407, −5.98193167556072136867214948558, −4.68962910008471583676586817804, −3.96124038740762337546166879602, −3.61696748763668538276811471075, −3.18116529021681048075075048091, −1.66339901487436461096988774561, 0,
1.66339901487436461096988774561, 3.18116529021681048075075048091, 3.61696748763668538276811471075, 3.96124038740762337546166879602, 4.68962910008471583676586817804, 5.98193167556072136867214948558, 6.61735010542873618183438048407, 6.90792465486007281618118516960, 8.245197862632869505068991524883
