| L(s) = 1 | + 2.69·3-s − 3.39·7-s + 4.28·9-s − 5.39·11-s + 1.28·13-s − 5.17·17-s − 1.77·19-s − 9.17·21-s − 23-s + 3.47·27-s − 4.36·29-s − 8.81·31-s − 14.5·33-s + 3.39·37-s + 3.47·39-s + 11.0·41-s − 6.57·43-s + 1.66·47-s + 4.55·49-s − 13.9·51-s + 0.954·53-s − 4.79·57-s + 8.97·59-s + 0.575·61-s − 14.5·63-s − 4.73·67-s − 2.69·69-s + ⋯ |
| L(s) = 1 | + 1.55·3-s − 1.28·7-s + 1.42·9-s − 1.62·11-s + 0.357·13-s − 1.25·17-s − 0.407·19-s − 2.00·21-s − 0.208·23-s + 0.669·27-s − 0.810·29-s − 1.58·31-s − 2.53·33-s + 0.558·37-s + 0.556·39-s + 1.72·41-s − 1.00·43-s + 0.243·47-s + 0.650·49-s − 1.95·51-s + 0.131·53-s − 0.635·57-s + 1.16·59-s + 0.0737·61-s − 1.83·63-s − 0.578·67-s − 0.324·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.69T + 3T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 + 5.39T + 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 + 5.17T + 17T^{2} \) |
| 19 | \( 1 + 1.77T + 19T^{2} \) |
| 29 | \( 1 + 4.36T + 29T^{2} \) |
| 31 | \( 1 + 8.81T + 31T^{2} \) |
| 37 | \( 1 - 3.39T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 - 1.66T + 47T^{2} \) |
| 53 | \( 1 - 0.954T + 53T^{2} \) |
| 59 | \( 1 - 8.97T + 59T^{2} \) |
| 61 | \( 1 - 0.575T + 61T^{2} \) |
| 67 | \( 1 + 4.73T + 67T^{2} \) |
| 71 | \( 1 + 9.88T + 71T^{2} \) |
| 73 | \( 1 - 3.18T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 1.17T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.836466111784042425046886926261, −7.83711637865878757427796665337, −7.35976052358875308384113183537, −6.40797774396680587911651123073, −5.50444494512280510922312128673, −4.27229827307947312684433557941, −3.50558641034007953980630349557, −2.70178882735733829876154172392, −2.08054658192431456112217589375, 0,
2.08054658192431456112217589375, 2.70178882735733829876154172392, 3.50558641034007953980630349557, 4.27229827307947312684433557941, 5.50444494512280510922312128673, 6.40797774396680587911651123073, 7.35976052358875308384113183537, 7.83711637865878757427796665337, 8.836466111784042425046886926261
