| L(s) = 1 | − 3-s + 4·7-s + 9-s − 4·11-s − 2·13-s + 2·17-s − 19-s − 4·21-s − 8·23-s − 27-s − 6·29-s − 4·31-s + 4·33-s − 10·37-s + 2·39-s − 2·41-s − 12·43-s + 9·49-s − 2·51-s + 6·53-s + 57-s + 10·61-s + 4·63-s + 4·67-s + 8·69-s + 8·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.229·19-s − 0.872·21-s − 1.66·23-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.696·33-s − 1.64·37-s + 0.320·39-s − 0.312·41-s − 1.82·43-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.132·57-s + 1.28·61-s + 0.503·63-s + 0.488·67-s + 0.963·69-s + 0.949·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8132805874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8132805874\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.94748071972400, −13.36218605272442, −12.78628084169949, −12.31926740719565, −11.76757862307006, −11.49882040725796, −10.88369773266525, −10.40527285456776, −10.07963729156173, −9.517278323061723, −8.599084925988031, −8.343614004768188, −7.688185090823154, −7.471020328329105, −6.779925631467063, −6.060986808925185, −5.387754157312633, −5.130528405710790, −4.792432888214581, −3.826414509094844, −3.539805794995664, −2.256032369300809, −2.092367160652571, −1.361592101909610, −0.2908209958959727,
0.2908209958959727, 1.361592101909610, 2.092367160652571, 2.256032369300809, 3.539805794995664, 3.826414509094844, 4.792432888214581, 5.130528405710790, 5.387754157312633, 6.060986808925185, 6.779925631467063, 7.471020328329105, 7.688185090823154, 8.343614004768188, 8.599084925988031, 9.517278323061723, 10.07963729156173, 10.40527285456776, 10.88369773266525, 11.49882040725796, 11.76757862307006, 12.31926740719565, 12.78628084169949, 13.36218605272442, 13.94748071972400
