| L(s) = 1 | − 3·3-s − 2·5-s − 4·7-s + 6·9-s + 2·11-s − 5·13-s + 6·15-s + 4·17-s − 2·19-s + 12·21-s + 23-s − 25-s − 9·27-s − 7·29-s − 3·31-s − 6·33-s + 8·35-s + 2·37-s + 15·39-s − 9·41-s − 8·43-s − 12·45-s + 9·47-s + 9·49-s − 12·51-s + 2·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894·5-s − 1.51·7-s + 2·9-s + 0.603·11-s − 1.38·13-s + 1.54·15-s + 0.970·17-s − 0.458·19-s + 2.61·21-s + 0.208·23-s − 1/5·25-s − 1.73·27-s − 1.29·29-s − 0.538·31-s − 1.04·33-s + 1.35·35-s + 0.328·37-s + 2.40·39-s − 1.40·41-s − 1.21·43-s − 1.78·45-s + 1.31·47-s + 9/7·49-s − 1.68·51-s + 0.274·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.03943744344058701436939538545, −12.21704194536701599614180308969, −11.67692432657301101982140211332, −10.38750929477444084894724644027, −9.520156045863341826363538518982, −7.41247065200411840033758437727, −6.53483233561451653204235369889, −5.32154529144374630720020311491, −3.80206517892802911800653447979, 0,
3.80206517892802911800653447979, 5.32154529144374630720020311491, 6.53483233561451653204235369889, 7.41247065200411840033758437727, 9.520156045863341826363538518982, 10.38750929477444084894724644027, 11.67692432657301101982140211332, 12.21704194536701599614180308969, 13.03943744344058701436939538545
