| L(s) = 1 | − 3-s + 9-s − 5·11-s + 2·13-s − 4·17-s + 6·19-s − 23-s − 27-s − 29-s − 2·31-s + 5·33-s − 5·37-s − 2·39-s + 7·43-s + 2·47-s + 4·51-s + 6·53-s − 6·57-s − 6·59-s − 3·67-s + 69-s + 9·71-s + 12·73-s + 3·79-s + 81-s + 6·83-s + 87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.50·11-s + 0.554·13-s − 0.970·17-s + 1.37·19-s − 0.208·23-s − 0.192·27-s − 0.185·29-s − 0.359·31-s + 0.870·33-s − 0.821·37-s − 0.320·39-s + 1.06·43-s + 0.291·47-s + 0.560·51-s + 0.824·53-s − 0.794·57-s − 0.781·59-s − 0.366·67-s + 0.120·69-s + 1.06·71-s + 1.40·73-s + 0.337·79-s + 1/9·81-s + 0.658·83-s + 0.107·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.10494007036109, −12.71631904344706, −12.14747585234133, −11.81599482772740, −11.15804685013319, −10.77775695379118, −10.58773351375971, −9.945880273046463, −9.373193297364145, −9.083581251239733, −8.290708825708693, −8.004758780336222, −7.380068858387797, −7.053080599785286, −6.444852999568042, −5.818890597076669, −5.490951788512035, −4.997814213845216, −4.551457670811806, −3.787775900867943, −3.391822155767759, −2.584445964704502, −2.209436740215732, −1.392187601564479, −0.6915725880160529, 0,
0.6915725880160529, 1.392187601564479, 2.209436740215732, 2.584445964704502, 3.391822155767759, 3.787775900867943, 4.551457670811806, 4.997814213845216, 5.490951788512035, 5.818890597076669, 6.444852999568042, 7.053080599785286, 7.380068858387797, 8.004758780336222, 8.290708825708693, 9.083581251239733, 9.373193297364145, 9.945880273046463, 10.58773351375971, 10.77775695379118, 11.15804685013319, 11.81599482772740, 12.14747585234133, 12.71631904344706, 13.10494007036109
