| L(s) = 1 | − 1.76·2-s − 1.09·3-s + 1.11·4-s + 1.93·6-s − 7-s + 1.56·8-s − 1.80·9-s − 0.239·11-s − 1.21·12-s + 1.56·13-s + 1.76·14-s − 4.98·16-s + 2.69·17-s + 3.17·18-s − 5.14·19-s + 1.09·21-s + 0.422·22-s + 23-s − 1.71·24-s − 2.75·26-s + 5.25·27-s − 1.11·28-s − 3.60·29-s + 6.22·31-s + 5.66·32-s + 0.262·33-s − 4.75·34-s + ⋯ |
| L(s) = 1 | − 1.24·2-s − 0.632·3-s + 0.555·4-s + 0.788·6-s − 0.377·7-s + 0.553·8-s − 0.600·9-s − 0.0721·11-s − 0.351·12-s + 0.433·13-s + 0.471·14-s − 1.24·16-s + 0.653·17-s + 0.748·18-s − 1.17·19-s + 0.238·21-s + 0.0900·22-s + 0.208·23-s − 0.350·24-s − 0.540·26-s + 1.01·27-s − 0.210·28-s − 0.668·29-s + 1.11·31-s + 1.00·32-s + 0.0456·33-s − 0.815·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.76T + 2T^{2} \) |
| 3 | \( 1 + 1.09T + 3T^{2} \) |
| 11 | \( 1 + 0.239T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 + 5.14T + 19T^{2} \) |
| 29 | \( 1 + 3.60T + 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 + 3.81T + 37T^{2} \) |
| 41 | \( 1 - 0.415T + 41T^{2} \) |
| 43 | \( 1 - 9.10T + 43T^{2} \) |
| 47 | \( 1 + 1.83T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 6.75T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 4.71T + 71T^{2} \) |
| 73 | \( 1 - 0.958T + 73T^{2} \) |
| 79 | \( 1 - 5.66T + 79T^{2} \) |
| 83 | \( 1 - 9.94T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 3.79T + 97T^{2} \) |
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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
−8.131895320262497686152543063866, −7.60830903570630128963790276983, −6.56975847522866793761966708219, −6.13914280959737852284247162036, −5.17832993125281601236338834480, −4.35469133624155081756528462656, −3.26854024670764544360277314167, −2.17411435960898659446090481149, −0.999168882474246751988793666800, 0,
0.999168882474246751988793666800, 2.17411435960898659446090481149, 3.26854024670764544360277314167, 4.35469133624155081756528462656, 5.17832993125281601236338834480, 6.13914280959737852284247162036, 6.56975847522866793761966708219, 7.60830903570630128963790276983, 8.131895320262497686152543063866
