| L(s) = 1 | + 2-s − 1.33·3-s + 4-s + 0.425·5-s − 1.33·6-s − 2.10·7-s + 8-s − 1.21·9-s + 0.425·10-s + 3.38·11-s − 1.33·12-s − 3.49·13-s − 2.10·14-s − 0.567·15-s + 16-s − 7.70·17-s − 1.21·18-s + 19-s + 0.425·20-s + 2.81·21-s + 3.38·22-s + 5.14·23-s − 1.33·24-s − 4.81·25-s − 3.49·26-s + 5.63·27-s − 2.10·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.770·3-s + 0.5·4-s + 0.190·5-s − 0.545·6-s − 0.797·7-s + 0.353·8-s − 0.405·9-s + 0.134·10-s + 1.01·11-s − 0.385·12-s − 0.970·13-s − 0.563·14-s − 0.146·15-s + 0.250·16-s − 1.86·17-s − 0.286·18-s + 0.229·19-s + 0.0950·20-s + 0.614·21-s + 0.720·22-s + 1.07·23-s − 0.272·24-s − 0.963·25-s − 0.686·26-s + 1.08·27-s − 0.398·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.590286597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.590286597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 - 0.425T + 5T^{2} \) |
| 7 | \( 1 + 2.10T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 + 7.70T + 17T^{2} \) |
| 23 | \( 1 - 5.14T + 23T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 + 0.0123T + 41T^{2} \) |
| 43 | \( 1 - 2.25T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 - 5.28T + 53T^{2} \) |
| 59 | \( 1 + 7.71T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 7.81T + 73T^{2} \) |
| 79 | \( 1 + 0.972T + 79T^{2} \) |
| 83 | \( 1 + 6.08T + 83T^{2} \) |
| 89 | \( 1 - 2.97T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−7.50188116156502750878522963706, −6.86689621001401792810483287332, −6.28314531554423030695077292213, −5.93063093160983381513394496925, −4.92805373317840435020514736967, −4.50973120431347533602866915292, −3.56796836867594364953171375740, −2.75474643026079945590583417769, −1.94746240933107547450007832721, −0.55673263664826584526163186489,
0.55673263664826584526163186489, 1.94746240933107547450007832721, 2.75474643026079945590583417769, 3.56796836867594364953171375740, 4.50973120431347533602866915292, 4.92805373317840435020514736967, 5.93063093160983381513394496925, 6.28314531554423030695077292213, 6.86689621001401792810483287332, 7.50188116156502750878522963706
