| L(s) = 1 | − 1.63·2-s + 1.29·3-s + 0.661·4-s + 4.07·5-s − 2.11·6-s + 2.52·7-s + 2.18·8-s − 1.31·9-s − 6.65·10-s − 0.486·11-s + 0.858·12-s + 2.08·13-s − 4.11·14-s + 5.29·15-s − 4.88·16-s − 2.18·17-s + 2.14·18-s + 1.54·19-s + 2.69·20-s + 3.27·21-s + 0.793·22-s + 7.37·23-s + 2.83·24-s + 11.6·25-s − 3.40·26-s − 5.60·27-s + 1.66·28-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.749·3-s + 0.330·4-s + 1.82·5-s − 0.864·6-s + 0.952·7-s + 0.772·8-s − 0.438·9-s − 2.10·10-s − 0.146·11-s + 0.247·12-s + 0.578·13-s − 1.09·14-s + 1.36·15-s − 1.22·16-s − 0.528·17-s + 0.505·18-s + 0.354·19-s + 0.602·20-s + 0.713·21-s + 0.169·22-s + 1.53·23-s + 0.578·24-s + 2.32·25-s − 0.666·26-s − 1.07·27-s + 0.315·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.757483134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.757483134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 \) |
| good | 2 | \( 1 + 1.63T + 2T^{2} \) |
| 3 | \( 1 - 1.29T + 3T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 + 0.486T + 11T^{2} \) |
| 13 | \( 1 - 2.08T + 13T^{2} \) |
| 17 | \( 1 + 2.18T + 17T^{2} \) |
| 19 | \( 1 - 1.54T + 19T^{2} \) |
| 23 | \( 1 - 7.37T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 + 1.14T + 31T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 4.08T + 43T^{2} \) |
| 47 | \( 1 + 4.86T + 47T^{2} \) |
| 53 | \( 1 - 9.12T + 53T^{2} \) |
| 59 | \( 1 + 9.77T + 59T^{2} \) |
| 61 | \( 1 + 3.99T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 + 4.97T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 1.23T + 83T^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| 97 | \( 1 + 5.37T + 97T^{2} \) |
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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
−9.392685155394539158803699304358, −8.753946044500634070455821784185, −8.466860327927329549440419300732, −7.40133653347110221032265227950, −6.47930951486230388514698130992, −5.42109202597658391983246595237, −4.70533848241891722358195697819, −3.02081278167604880995327249150, −2.03273214146964295602233158128, −1.26047768118436227322104932205,
1.26047768118436227322104932205, 2.03273214146964295602233158128, 3.02081278167604880995327249150, 4.70533848241891722358195697819, 5.42109202597658391983246595237, 6.47930951486230388514698130992, 7.40133653347110221032265227950, 8.466860327927329549440419300732, 8.753946044500634070455821784185, 9.392685155394539158803699304358
