L(s) = 1 | − 3-s + 7-s − 2·9-s + 3·11-s + 3·13-s + 3·17-s − 6·19-s − 21-s + 23-s + 5·27-s + 3·29-s + 4·31-s − 3·33-s − 8·37-s − 3·39-s − 6·41-s − 8·43-s + 3·47-s + 49-s − 3·51-s − 4·53-s + 6·57-s − 4·59-s + 6·61-s − 2·63-s + 6·67-s − 69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.832·13-s + 0.727·17-s − 1.37·19-s − 0.218·21-s + 0.208·23-s + 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.522·33-s − 1.31·37-s − 0.480·39-s − 0.937·41-s − 1.21·43-s + 0.437·47-s + 1/7·49-s − 0.420·51-s − 0.549·53-s + 0.794·57-s − 0.520·59-s + 0.768·61-s − 0.251·63-s + 0.733·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.46557127841916, −14.00108549304876, −13.59494589106176, −12.89727323942525, −12.33795689827836, −11.88940201564957, −11.53439942016427, −10.97728757592972, −10.49910999690386, −10.09078223994567, −9.305016794212162, −8.722961078373294, −8.359574887759659, −7.986027771446374, −6.935826601938189, −6.612154482442829, −6.170163915854339, −5.518830311930517, −5.003606986458444, −4.414499781221011, −3.674767864856756, −3.235970555669857, −2.338625970047396, −1.577547116552242, −0.9416092976864130, 0,
0.9416092976864130, 1.577547116552242, 2.338625970047396, 3.235970555669857, 3.674767864856756, 4.414499781221011, 5.003606986458444, 5.518830311930517, 6.170163915854339, 6.612154482442829, 6.935826601938189, 7.986027771446374, 8.359574887759659, 8.722961078373294, 9.305016794212162, 10.09078223994567, 10.49910999690386, 10.97728757592972, 11.53439942016427, 11.88940201564957, 12.33795689827836, 12.89727323942525, 13.59494589106176, 14.00108549304876, 14.46557127841916