| L(s) = 1 | + 2-s + 0.745·3-s + 4-s − 3.79·5-s + 0.745·6-s − 4.73·7-s + 8-s − 2.44·9-s − 3.79·10-s + 0.848·11-s + 0.745·12-s + 4.15·13-s − 4.73·14-s − 2.82·15-s + 16-s − 1.02·17-s − 2.44·18-s − 1.39·19-s − 3.79·20-s − 3.52·21-s + 0.848·22-s − 2.80·23-s + 0.745·24-s + 9.40·25-s + 4.15·26-s − 4.05·27-s − 4.73·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.430·3-s + 0.5·4-s − 1.69·5-s + 0.304·6-s − 1.79·7-s + 0.353·8-s − 0.814·9-s − 1.20·10-s + 0.255·11-s + 0.215·12-s + 1.15·13-s − 1.26·14-s − 0.730·15-s + 0.250·16-s − 0.249·17-s − 0.576·18-s − 0.320·19-s − 0.848·20-s − 0.770·21-s + 0.180·22-s − 0.585·23-s + 0.152·24-s + 1.88·25-s + 0.815·26-s − 0.780·27-s − 0.895·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.391722747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391722747\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 0.745T + 3T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 0.848T + 11T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 + 1.02T + 17T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 23 | \( 1 + 2.80T + 23T^{2} \) |
| 29 | \( 1 + 5.22T + 29T^{2} \) |
| 31 | \( 1 - 2.16T + 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 41 | \( 1 + 0.954T + 41T^{2} \) |
| 43 | \( 1 - 2.36T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 3.14T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 9.19T + 71T^{2} \) |
| 73 | \( 1 + 0.665T + 73T^{2} \) |
| 79 | \( 1 - 7.75T + 79T^{2} \) |
| 83 | \( 1 - 5.40T + 83T^{2} \) |
| 89 | \( 1 + 7.63T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.269292232508504813945473617993, −7.80272895289720644393482802321, −6.77598691613717907828873203989, −6.36546070627374752479884507195, −5.55018548746907972257197121979, −4.28553694290182241621509618378, −3.64597202594317772264310147423, −3.37214130339464361469762069157, −2.43195683672415307245569182960, −0.56163496944823265311944922882,
0.56163496944823265311944922882, 2.43195683672415307245569182960, 3.37214130339464361469762069157, 3.64597202594317772264310147423, 4.28553694290182241621509618378, 5.55018548746907972257197121979, 6.36546070627374752479884507195, 6.77598691613717907828873203989, 7.80272895289720644393482802321, 8.269292232508504813945473617993
