| L(s) = 1 | + 2.16·2-s + 2.91·3-s + 2.67·4-s − 2.73·5-s + 6.31·6-s + 2.02·7-s + 1.46·8-s + 5.51·9-s − 5.90·10-s − 0.826·11-s + 7.81·12-s + 3.16·13-s + 4.37·14-s − 7.96·15-s − 2.18·16-s + 3.41·17-s + 11.9·18-s + 2.96·19-s − 7.31·20-s + 5.89·21-s − 1.78·22-s − 1.22·23-s + 4.28·24-s + 2.45·25-s + 6.83·26-s + 7.33·27-s + 5.41·28-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.68·3-s + 1.33·4-s − 1.22·5-s + 2.57·6-s + 0.764·7-s + 0.518·8-s + 1.83·9-s − 1.86·10-s − 0.249·11-s + 2.25·12-s + 0.876·13-s + 1.16·14-s − 2.05·15-s − 0.545·16-s + 0.828·17-s + 2.81·18-s + 0.679·19-s − 1.63·20-s + 1.28·21-s − 0.381·22-s − 0.255·23-s + 0.874·24-s + 0.490·25-s + 1.34·26-s + 1.41·27-s + 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.435907869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.435907869\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 2.16T + 2T^{2} \) |
| 3 | \( 1 - 2.91T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 - 2.02T + 7T^{2} \) |
| 11 | \( 1 + 0.826T + 11T^{2} \) |
| 13 | \( 1 - 3.16T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 - 2.96T + 19T^{2} \) |
| 23 | \( 1 + 1.22T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 5.33T + 31T^{2} \) |
| 37 | \( 1 + 3.46T + 37T^{2} \) |
| 41 | \( 1 + 6.88T + 41T^{2} \) |
| 47 | \( 1 - 3.75T + 47T^{2} \) |
| 53 | \( 1 - 8.87T + 53T^{2} \) |
| 59 | \( 1 - 2.84T + 59T^{2} \) |
| 61 | \( 1 + 7.67T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 4.68T + 79T^{2} \) |
| 83 | \( 1 + 4.77T + 83T^{2} \) |
| 89 | \( 1 - 2.06T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.836790533517274503470520646446, −8.404245590108336023347396042790, −7.61762894063644417419882686189, −7.09651259814329818171419252125, −5.81409678116482298316238441171, −4.79273003459067622840134428808, −4.08588378281518238047271443277, −3.42766319299153730792906453479, −2.88063560347982576845931752380, −1.59283997021720135880941150673,
1.59283997021720135880941150673, 2.88063560347982576845931752380, 3.42766319299153730792906453479, 4.08588378281518238047271443277, 4.79273003459067622840134428808, 5.81409678116482298316238441171, 7.09651259814329818171419252125, 7.61762894063644417419882686189, 8.404245590108336023347396042790, 8.836790533517274503470520646446
