| L(s) = 1 | − 2.65·2-s + 0.121·3-s + 5.04·4-s − 1.06·5-s − 0.323·6-s + 3.36·7-s − 8.08·8-s − 2.98·9-s + 2.81·10-s + 0.614·12-s + 2.15·13-s − 8.94·14-s − 0.129·15-s + 11.3·16-s − 3.67·17-s + 7.92·18-s + 19-s − 5.34·20-s + 0.410·21-s + 3.15·23-s − 0.985·24-s − 3.87·25-s − 5.73·26-s − 0.729·27-s + 16.9·28-s − 7.17·29-s + 0.342·30-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.0703·3-s + 2.52·4-s − 0.474·5-s − 0.132·6-s + 1.27·7-s − 2.85·8-s − 0.995·9-s + 0.889·10-s + 0.177·12-s + 0.599·13-s − 2.38·14-s − 0.0333·15-s + 2.84·16-s − 0.890·17-s + 1.86·18-s + 0.229·19-s − 1.19·20-s + 0.0895·21-s + 0.657·23-s − 0.201·24-s − 0.775·25-s − 1.12·26-s − 0.140·27-s + 3.21·28-s − 1.33·29-s + 0.0626·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 - 0.121T + 3T^{2} \) |
| 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 - 3.36T + 7T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 17 | \( 1 + 3.67T + 17T^{2} \) |
| 23 | \( 1 - 3.15T + 23T^{2} \) |
| 29 | \( 1 + 7.17T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 - 2.27T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 9.38T + 43T^{2} \) |
| 47 | \( 1 + 5.77T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 6.98T + 61T^{2} \) |
| 67 | \( 1 + 4.81T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 8.08T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 9.96T + 83T^{2} \) |
| 89 | \( 1 + 4.61T + 89T^{2} \) |
| 97 | \( 1 + 4.09T + 97T^{2} \) |
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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
−8.676612694677512953083915068974, −7.87761473852313613871523503189, −7.67903966968954923079573981189, −6.54730362102695662247860691318, −5.80987793145193828417157843917, −4.64856379336519166452380844919, −3.30512605211977651883884992439, −2.25423917546517488283456584322, −1.34265527918121425302683720678, 0,
1.34265527918121425302683720678, 2.25423917546517488283456584322, 3.30512605211977651883884992439, 4.64856379336519166452380844919, 5.80987793145193828417157843917, 6.54730362102695662247860691318, 7.67903966968954923079573981189, 7.87761473852313613871523503189, 8.676612694677512953083915068974
