| L(s) = 1 | + 0.775·2-s − 3-s − 1.39·4-s − 2.32·5-s − 0.775·6-s − 2.63·8-s + 9-s − 1.80·10-s + 0.760·11-s + 1.39·12-s − 5.49·13-s + 2.32·15-s + 0.752·16-s − 0.343·17-s + 0.775·18-s − 4.11·19-s + 3.24·20-s + 0.589·22-s − 2.25·23-s + 2.63·24-s + 0.399·25-s − 4.25·26-s − 27-s − 2.50·29-s + 1.80·30-s − 9.19·31-s + 5.85·32-s + ⋯ |
| L(s) = 1 | + 0.548·2-s − 0.577·3-s − 0.699·4-s − 1.03·5-s − 0.316·6-s − 0.931·8-s + 0.333·9-s − 0.569·10-s + 0.229·11-s + 0.403·12-s − 1.52·13-s + 0.599·15-s + 0.188·16-s − 0.0832·17-s + 0.182·18-s − 0.943·19-s + 0.726·20-s + 0.125·22-s − 0.470·23-s + 0.538·24-s + 0.0799·25-s − 0.835·26-s − 0.192·27-s − 0.464·29-s + 0.329·30-s − 1.65·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1684805396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1684805396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 0.775T + 2T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 11 | \( 1 - 0.760T + 11T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 + 0.343T + 17T^{2} \) |
| 19 | \( 1 + 4.11T + 19T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 + 2.50T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 + 3.66T + 37T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 2.71T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 1.48T + 59T^{2} \) |
| 61 | \( 1 - 1.90T + 61T^{2} \) |
| 67 | \( 1 + 7.92T + 67T^{2} \) |
| 71 | \( 1 + 9.05T + 71T^{2} \) |
| 73 | \( 1 - 9.45T + 73T^{2} \) |
| 79 | \( 1 + 3.15T + 79T^{2} \) |
| 83 | \( 1 + 1.77T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.056902656797055181652016842370, −7.24906490824916635415648935117, −6.72300861728896390910933763681, −5.64370754479301514293197216483, −5.20182934075156603843261802492, −4.30918256353712158147321895323, −3.99024077629256095920606970311, −3.06536683232911723053215572642, −1.86842303893420165931777436065, −0.19609440339128153247876485776,
0.19609440339128153247876485776, 1.86842303893420165931777436065, 3.06536683232911723053215572642, 3.99024077629256095920606970311, 4.30918256353712158147321895323, 5.20182934075156603843261802492, 5.64370754479301514293197216483, 6.72300861728896390910933763681, 7.24906490824916635415648935117, 8.056902656797055181652016842370
