| L(s) = 1 | + 1.92·3-s − 2.25·5-s + 2.05·7-s + 0.698·9-s + 11-s − 0.869·13-s − 4.33·15-s + 2.41·17-s − 5.91·19-s + 3.95·21-s − 2.27·23-s + 0.0707·25-s − 4.42·27-s − 3.46·29-s − 2.48·31-s + 1.92·33-s − 4.63·35-s − 0.292·37-s − 1.67·39-s − 9.11·41-s + 2.13·43-s − 1.57·45-s + 3.45·47-s − 2.76·49-s + 4.63·51-s + 53-s − 2.25·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s − 1.00·5-s + 0.777·7-s + 0.232·9-s + 0.301·11-s − 0.241·13-s − 1.11·15-s + 0.584·17-s − 1.35·19-s + 0.863·21-s − 0.474·23-s + 0.0141·25-s − 0.851·27-s − 0.642·29-s − 0.445·31-s + 0.334·33-s − 0.782·35-s − 0.0480·37-s − 0.267·39-s − 1.42·41-s + 0.324·43-s − 0.234·45-s + 0.503·47-s − 0.395·49-s + 0.649·51-s + 0.137·53-s − 0.303·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 1.92T + 3T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 13 | \( 1 + 0.869T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 + 5.91T + 19T^{2} \) |
| 23 | \( 1 + 2.27T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 37 | \( 1 + 0.292T + 37T^{2} \) |
| 41 | \( 1 + 9.11T + 41T^{2} \) |
| 43 | \( 1 - 2.13T + 43T^{2} \) |
| 47 | \( 1 - 3.45T + 47T^{2} \) |
| 59 | \( 1 + 3.53T + 59T^{2} \) |
| 61 | \( 1 + 6.97T + 61T^{2} \) |
| 67 | \( 1 - 8.28T + 67T^{2} \) |
| 71 | \( 1 - 3.45T + 71T^{2} \) |
| 73 | \( 1 + 4.39T + 73T^{2} \) |
| 79 | \( 1 - 7.14T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 7.00T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.006017983178598752690044476425, −7.57022936223866496639761865894, −6.70051232316170273422136238584, −5.70455952919973152295666141345, −4.77004397090364859358046646528, −3.95173529476229758056804960104, −3.49690909530943251684758094161, −2.42394811565367632503121736157, −1.63004044294217811563473394270, 0,
1.63004044294217811563473394270, 2.42394811565367632503121736157, 3.49690909530943251684758094161, 3.95173529476229758056804960104, 4.77004397090364859358046646528, 5.70455952919973152295666141345, 6.70051232316170273422136238584, 7.57022936223866496639761865894, 8.006017983178598752690044476425
