| L(s) = 1 | + 7-s + 3.16·11-s + 3·13-s + 6.32·17-s + 3·19-s − 3.16·23-s − 9.48·29-s − 2·31-s + 37-s + 3.16·41-s + 10·43-s − 6.32·47-s − 6·49-s − 9.48·53-s + 6.32·59-s − 61-s + 11·67-s − 9.48·71-s + 13·73-s + 3.16·77-s − 3·79-s + 15.8·83-s + 12.6·89-s + 3·91-s − 97-s + 6.32·101-s + 17·103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.953·11-s + 0.832·13-s + 1.53·17-s + 0.688·19-s − 0.659·23-s − 1.76·29-s − 0.359·31-s + 0.164·37-s + 0.493·41-s + 1.52·43-s − 0.922·47-s − 0.857·49-s − 1.30·53-s + 0.823·59-s − 0.128·61-s + 1.34·67-s − 1.12·71-s + 1.52·73-s + 0.360·77-s − 0.337·79-s + 1.73·83-s + 1.34·89-s + 0.314·91-s − 0.101·97-s + 0.629·101-s + 1.67·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.232315277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.232315277\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + 9.48T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 3.16T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 - 6.32T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 + 9.48T + 71T^{2} \) |
| 73 | \( 1 - 13T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−8.952216281774163655149709599124, −7.87915716201208839759976042805, −7.57726448886583275414283355078, −6.42351093379022114849647595739, −5.80388534522964012995854624341, −5.00807081905842181684732879714, −3.85427619247924432272294062973, −3.39568141620960713915785210161, −1.93335904934305200892609432399, −1.00929272439739406199057095775,
1.00929272439739406199057095775, 1.93335904934305200892609432399, 3.39568141620960713915785210161, 3.85427619247924432272294062973, 5.00807081905842181684732879714, 5.80388534522964012995854624341, 6.42351093379022114849647595739, 7.57726448886583275414283355078, 7.87915716201208839759976042805, 8.952216281774163655149709599124
