| L(s) = 1 | − 3-s + 9-s + 4·11-s − 2·13-s + 6·17-s + 4·19-s − 23-s − 27-s + 2·29-s − 4·33-s − 2·37-s + 2·39-s + 10·41-s + 4·43-s − 7·49-s − 6·51-s + 6·53-s − 4·57-s − 4·59-s + 10·61-s + 12·67-s + 69-s + 8·71-s − 10·73-s + 8·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.208·23-s − 0.192·27-s + 0.371·29-s − 0.696·33-s − 0.328·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s − 49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 1.28·61-s + 1.46·67-s + 0.120·69-s + 0.949·71-s − 1.17·73-s + 0.900·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.981671310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.981671310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.87209283619507, −13.01924217970722, −12.50523452087473, −12.22002490739251, −11.69183196025963, −11.38232564290157, −10.77680315044335, −10.11830943423764, −9.782622355542153, −9.359481151157216, −8.810832958881002, −8.099534096228732, −7.536055594911184, −7.261585186493061, −6.459808587484391, −6.190861216931073, −5.430487541859308, −5.145086110850691, −4.427158712629028, −3.794372013944306, −3.384800048772382, −2.590526370951693, −1.845551953803700, −1.034329929540997, −0.6835701320061329,
0.6835701320061329, 1.034329929540997, 1.845551953803700, 2.590526370951693, 3.384800048772382, 3.794372013944306, 4.427158712629028, 5.145086110850691, 5.430487541859308, 6.190861216931073, 6.459808587484391, 7.261585186493061, 7.536055594911184, 8.099534096228732, 8.810832958881002, 9.359481151157216, 9.782622355542153, 10.11830943423764, 10.77680315044335, 11.38232564290157, 11.69183196025963, 12.22002490739251, 12.50523452087473, 13.01924217970722, 13.87209283619507
