| L(s) = 1 | + 2-s + 2.73·3-s + 4-s + 2.73·6-s + 1.26·7-s + 8-s + 4.46·9-s + 1.46·11-s + 2.73·12-s − 1.46·13-s + 1.26·14-s + 16-s + 1.46·17-s + 4.46·18-s − 4.19·19-s + 3.46·21-s + 1.46·22-s + 8·23-s + 2.73·24-s − 1.46·26-s + 3.99·27-s + 1.26·28-s − 8.92·29-s − 2.73·31-s + 32-s + 4·33-s + 1.46·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.57·3-s + 0.5·4-s + 1.11·6-s + 0.479·7-s + 0.353·8-s + 1.48·9-s + 0.441·11-s + 0.788·12-s − 0.406·13-s + 0.338·14-s + 0.250·16-s + 0.355·17-s + 1.05·18-s − 0.962·19-s + 0.755·21-s + 0.312·22-s + 1.66·23-s + 0.557·24-s − 0.287·26-s + 0.769·27-s + 0.239·28-s − 1.65·29-s − 0.490·31-s + 0.176·32-s + 0.696·33-s + 0.251·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.988401710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.988401710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 + 4.19T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 8.92T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 - 1.26T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 0.196T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 5.26T + 79T^{2} \) |
| 83 | \( 1 - 5.26T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.004001430377689944915927007538, −8.598733423047630184155344837903, −7.47499756616071603867196585268, −7.23218117195997713434059500337, −5.99270558720571648066210713444, −4.95707356291623409317770773447, −4.09544556474249645892883802569, −3.34847261523396353140969350432, −2.45575591700564970785754356198, −1.57118002568879529537407800860,
1.57118002568879529537407800860, 2.45575591700564970785754356198, 3.34847261523396353140969350432, 4.09544556474249645892883802569, 4.95707356291623409317770773447, 5.99270558720571648066210713444, 7.23218117195997713434059500337, 7.47499756616071603867196585268, 8.598733423047630184155344837903, 9.004001430377689944915927007538
