| L(s) = 1 | − 3-s + 9-s − 13-s + 2·17-s − 3·19-s − 6·23-s − 27-s − 6·29-s − 7·31-s − 37-s + 39-s − 4·41-s + 11·43-s + 6·47-s − 2·51-s + 4·53-s + 3·57-s + 14·59-s + 6·61-s − 3·67-s + 6·69-s + 7·73-s + 9·79-s + 81-s + 16·83-s + 6·87-s + 2·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.277·13-s + 0.485·17-s − 0.688·19-s − 1.25·23-s − 0.192·27-s − 1.11·29-s − 1.25·31-s − 0.164·37-s + 0.160·39-s − 0.624·41-s + 1.67·43-s + 0.875·47-s − 0.280·51-s + 0.549·53-s + 0.397·57-s + 1.82·59-s + 0.768·61-s − 0.366·67-s + 0.722·69-s + 0.819·73-s + 1.01·79-s + 1/9·81-s + 1.75·83-s + 0.643·87-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.45228727962394, −14.79968311047857, −14.52587418077337, −13.80731627344564, −13.23614122192296, −12.70996325919118, −12.17590597333553, −11.80977396563180, −11.05531666700164, −10.67744900262783, −10.11907376728482, −9.498077411179012, −9.018901641095560, −8.261329557433345, −7.676726361742719, −7.164995991329135, −6.540571100175792, −5.788222668153428, −5.500340388017794, −4.761880241096945, −3.852322275122410, −3.708058427007419, −2.396746380132291, −1.977887626256961, −0.9013643959904470, 0,
0.9013643959904470, 1.977887626256961, 2.396746380132291, 3.708058427007419, 3.852322275122410, 4.761880241096945, 5.500340388017794, 5.788222668153428, 6.540571100175792, 7.164995991329135, 7.676726361742719, 8.261329557433345, 9.018901641095560, 9.498077411179012, 10.11907376728482, 10.67744900262783, 11.05531666700164, 11.80977396563180, 12.17590597333553, 12.70996325919118, 13.23614122192296, 13.80731627344564, 14.52587418077337, 14.79968311047857, 15.45228727962394
