| L(s) = 1 | − 3-s + 9-s − 2·13-s − 2·17-s − 6·19-s + 6·23-s − 27-s − 2·29-s − 2·31-s − 2·37-s + 2·39-s + 2·41-s + 8·43-s + 2·51-s + 6·57-s + 8·67-s − 6·69-s + 10·73-s − 8·79-s + 81-s − 12·83-s + 2·87-s − 6·89-s + 2·93-s + 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.554·13-s − 0.485·17-s − 1.37·19-s + 1.25·23-s − 0.192·27-s − 0.371·29-s − 0.359·31-s − 0.328·37-s + 0.320·39-s + 0.312·41-s + 1.21·43-s + 0.280·51-s + 0.794·57-s + 0.977·67-s − 0.722·69-s + 1.17·73-s − 0.900·79-s + 1/9·81-s − 1.31·83-s + 0.214·87-s − 0.635·89-s + 0.207·93-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-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 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.47631132282305, −14.96116482293441, −14.39890308923198, −13.90171743770298, −13.06017328851610, −12.72782197247316, −12.44299583836325, −11.46849729428798, −11.25344459776178, −10.63328914628511, −10.18046068682561, −9.447381691533054, −8.945624656224861, −8.413366637617731, −7.617242108772611, −7.077403770491768, −6.601972609916160, −5.927642425238856, −5.351722537195738, −4.662220769984015, −4.212211854607302, −3.401967460514762, −2.517242461164176, −1.923933876116534, −0.8987335885612974, 0,
0.8987335885612974, 1.923933876116534, 2.517242461164176, 3.401967460514762, 4.212211854607302, 4.662220769984015, 5.351722537195738, 5.927642425238856, 6.601972609916160, 7.077403770491768, 7.617242108772611, 8.413366637617731, 8.945624656224861, 9.447381691533054, 10.18046068682561, 10.63328914628511, 11.25344459776178, 11.46849729428798, 12.44299583836325, 12.72782197247316, 13.06017328851610, 13.90171743770298, 14.39890308923198, 14.96116482293441, 15.47631132282305
