| L(s) = 1 | − 2·7-s − 4.47·11-s − 4.47·13-s − 4.47·17-s + 4·23-s − 4·29-s − 8.94·31-s + 4.47·37-s − 10·41-s + 4·43-s + 8·47-s − 3·49-s − 4.47·53-s + 13.4·59-s + 10·61-s + 8·67-s − 8.94·71-s + 8.94·73-s + 8.94·77-s + 8.94·79-s − 4·83-s − 6·89-s + 8.94·91-s − 17.8·97-s − 12·101-s + 14·103-s − 12·107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.34·11-s − 1.24·13-s − 1.08·17-s + 0.834·23-s − 0.742·29-s − 1.60·31-s + 0.735·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 0.428·49-s − 0.614·53-s + 1.74·59-s + 1.28·61-s + 0.977·67-s − 1.06·71-s + 1.04·73-s + 1.01·77-s + 1.00·79-s − 0.439·83-s − 0.635·89-s + 0.937·91-s − 1.81·97-s − 1.19·101-s + 1.37·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7333592384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7333592384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 4.47T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 8.94T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−7.86871991573325224939348751896, −7.07024958420096836436761732147, −6.81139227950794083438796750640, −5.61183421286520685307648358152, −5.24556762401503450103670387437, −4.39134618792322913368759335981, −3.47822808211734749717777637256, −2.63175445203590428690369700642, −2.05837628116260045019604295053, −0.40135572188628562551354546155,
0.40135572188628562551354546155, 2.05837628116260045019604295053, 2.63175445203590428690369700642, 3.47822808211734749717777637256, 4.39134618792322913368759335981, 5.24556762401503450103670387437, 5.61183421286520685307648358152, 6.81139227950794083438796750640, 7.07024958420096836436761732147, 7.86871991573325224939348751896
