| L(s) = 1 | − 3.28·3-s + 1.93·7-s + 7.80·9-s − 5.62·11-s − 2.07·13-s + 3.42·17-s − 19-s − 6.35·21-s + 5.35·23-s − 15.7·27-s + 1.09·29-s − 3.09·31-s + 18.5·33-s + 3.28·37-s + 6.80·39-s − 11.6·41-s + 0.501·43-s + 12.6·47-s − 3.26·49-s − 11.2·51-s + 3.01·53-s + 3.28·57-s + 2.35·59-s + 7.62·61-s + 15.0·63-s − 12.2·67-s − 17.6·69-s + ⋯ |
| L(s) = 1 | − 1.89·3-s + 0.730·7-s + 2.60·9-s − 1.69·11-s − 0.574·13-s + 0.830·17-s − 0.229·19-s − 1.38·21-s + 1.11·23-s − 3.03·27-s + 0.202·29-s − 0.555·31-s + 3.22·33-s + 0.540·37-s + 1.08·39-s − 1.81·41-s + 0.0764·43-s + 1.84·47-s − 0.466·49-s − 1.57·51-s + 0.413·53-s + 0.435·57-s + 0.306·59-s + 0.976·61-s + 1.89·63-s − 1.49·67-s − 2.12·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3.28T + 3T^{2} \) |
| 7 | \( 1 - 1.93T + 7T^{2} \) |
| 11 | \( 1 + 5.62T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 23 | \( 1 - 5.35T + 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 - 3.28T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 0.501T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 3.01T + 53T^{2} \) |
| 59 | \( 1 - 2.35T + 59T^{2} \) |
| 61 | \( 1 - 7.62T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 5.35T + 83T^{2} \) |
| 89 | \( 1 - 8.35T + 89T^{2} \) |
| 97 | \( 1 - 3.01T + 97T^{2} \) |
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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
−7.34861614357834712840652333652, −6.91321169047358639124134827972, −5.79585879666034393519436593310, −5.50149850453353616416249126887, −4.87313170240053531138948037090, −4.40328245440857404593036108764, −3.12638530428873908343045023745, −2.01254104507312740418965229795, −0.990316997852906773524439996582, 0,
0.990316997852906773524439996582, 2.01254104507312740418965229795, 3.12638530428873908343045023745, 4.40328245440857404593036108764, 4.87313170240053531138948037090, 5.50149850453353616416249126887, 5.79585879666034393519436593310, 6.91321169047358639124134827972, 7.34861614357834712840652333652
