| L(s) = 1 | + 1.38·2-s − 2.82·3-s − 0.0791·4-s + 0.518·5-s − 3.91·6-s + 7-s − 2.88·8-s + 4.98·9-s + 0.719·10-s + 1.62·11-s + 0.223·12-s + 1.38·14-s − 1.46·15-s − 3.83·16-s + 1.94·17-s + 6.90·18-s + 2.49·19-s − 0.0410·20-s − 2.82·21-s + 2.25·22-s − 9.14·23-s + 8.14·24-s − 4.73·25-s − 5.60·27-s − 0.0791·28-s − 5.22·29-s − 2.03·30-s + ⋯ |
| L(s) = 1 | + 0.980·2-s − 1.63·3-s − 0.0395·4-s + 0.232·5-s − 1.59·6-s + 0.377·7-s − 1.01·8-s + 1.66·9-s + 0.227·10-s + 0.489·11-s + 0.0645·12-s + 0.370·14-s − 0.378·15-s − 0.958·16-s + 0.472·17-s + 1.62·18-s + 0.571·19-s − 0.00918·20-s − 0.616·21-s + 0.479·22-s − 1.90·23-s + 1.66·24-s − 0.946·25-s − 1.07·27-s − 0.0149·28-s − 0.971·29-s − 0.371·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.38T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 0.518T + 5T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 + 9.14T + 23T^{2} \) |
| 29 | \( 1 + 5.22T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 + 0.997T + 43T^{2} \) |
| 47 | \( 1 + 4.51T + 47T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 - 6.20T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 8.37T + 67T^{2} \) |
| 71 | \( 1 + 5.19T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 0.982T + 79T^{2} \) |
| 83 | \( 1 + 8.91T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−9.725758991492974946035634815996, −8.514761154098642350301655971054, −7.40572611373436343388521444653, −6.31265196311506045712708155121, −5.86384833724499369807895839009, −5.14893702181746207585536562169, −4.39125987397909056111492520928, −3.49024700644794349302395497842, −1.67943871679761165427042999432, 0,
1.67943871679761165427042999432, 3.49024700644794349302395497842, 4.39125987397909056111492520928, 5.14893702181746207585536562169, 5.86384833724499369807895839009, 6.31265196311506045712708155121, 7.40572611373436343388521444653, 8.514761154098642350301655971054, 9.725758991492974946035634815996
