| L(s) = 1 | − 3-s − 2·4-s + 2·5-s + 3·7-s + 9-s + 2·12-s − 2·15-s + 4·16-s − 2·17-s − 4·19-s − 4·20-s − 3·21-s − 4·23-s − 25-s − 27-s − 6·28-s + 5·31-s + 6·35-s − 2·36-s − 6·37-s + 8·41-s − 5·43-s + 2·45-s − 4·48-s + 2·49-s + 2·51-s + 14·53-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.894·5-s + 1.13·7-s + 1/3·9-s + 0.577·12-s − 0.516·15-s + 16-s − 0.485·17-s − 0.917·19-s − 0.894·20-s − 0.654·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.13·28-s + 0.898·31-s + 1.01·35-s − 1/3·36-s − 0.986·37-s + 1.24·41-s − 0.762·43-s + 0.298·45-s − 0.577·48-s + 2/7·49-s + 0.280·51-s + 1.92·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.696642643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.696642643\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.17747854641243, −13.75832809260414, −13.38572569186441, −12.86712871664902, −12.15698729217966, −11.95769938894245, −11.04247799302750, −10.82567006912828, −10.07686334011575, −9.831217509648374, −9.170205214705769, −8.582482694977146, −8.171570660472503, −7.702336780831186, −6.763372402099466, −6.380194843986389, −5.612677584663898, −5.299143531432301, −4.770839785392275, −4.112125003860808, −3.776554494218241, −2.511989293705176, −2.010738085895455, −1.293530266760436, −0.4911029033852267,
0.4911029033852267, 1.293530266760436, 2.010738085895455, 2.511989293705176, 3.776554494218241, 4.112125003860808, 4.770839785392275, 5.299143531432301, 5.612677584663898, 6.380194843986389, 6.763372402099466, 7.702336780831186, 8.171570660472503, 8.582482694977146, 9.170205214705769, 9.831217509648374, 10.07686334011575, 10.82567006912828, 11.04247799302750, 11.95769938894245, 12.15698729217966, 12.86712871664902, 13.38572569186441, 13.75832809260414, 14.17747854641243
