| L(s) = 1 | − 0.517·2-s − 1.73·4-s − 2.17·7-s + 1.93·8-s − 3.07·11-s + 0.582·13-s + 1.12·14-s + 2.46·16-s − 1.41·17-s − 2·19-s + 1.59·22-s + 1.41·23-s − 0.301·26-s + 3.76·28-s + 7.27·29-s + 31-s − 5.13·32-s + 0.732·34-s + 5.36·37-s + 1.03·38-s + 0.824·41-s − 1.59·43-s + 5.32·44-s − 0.732·46-s − 3.20·47-s − 2.26·49-s − 1.00·52-s + ⋯ |
| L(s) = 1 | − 0.366·2-s − 0.866·4-s − 0.822·7-s + 0.683·8-s − 0.927·11-s + 0.161·13-s + 0.300·14-s + 0.616·16-s − 0.342·17-s − 0.458·19-s + 0.339·22-s + 0.294·23-s − 0.0591·26-s + 0.712·28-s + 1.35·29-s + 0.179·31-s − 0.908·32-s + 0.125·34-s + 0.881·37-s + 0.167·38-s + 0.128·41-s − 0.242·43-s + 0.803·44-s − 0.107·46-s − 0.467·47-s − 0.323·49-s − 0.140·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.517T + 2T^{2} \) |
| 7 | \( 1 + 2.17T + 7T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 - 0.582T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 7.27T + 29T^{2} \) |
| 37 | \( 1 - 5.36T + 37T^{2} \) |
| 41 | \( 1 - 0.824T + 41T^{2} \) |
| 43 | \( 1 + 1.59T + 43T^{2} \) |
| 47 | \( 1 + 3.20T + 47T^{2} \) |
| 53 | \( 1 + 0.378T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 8.10T + 71T^{2} \) |
| 73 | \( 1 - 5.36T + 73T^{2} \) |
| 79 | \( 1 - 0.196T + 79T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 + 5.85T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.85982093784035209264751743846, −6.86816901578023874043324294053, −6.30691811549907882606758936220, −5.37930932221440899458725001800, −4.77035896001698335961676140366, −4.00723050102350484543656615005, −3.15999953301909399362168505594, −2.32873460837281239652774197895, −0.985920639360519975501905941584, 0,
0.985920639360519975501905941584, 2.32873460837281239652774197895, 3.15999953301909399362168505594, 4.00723050102350484543656615005, 4.77035896001698335961676140366, 5.37930932221440899458725001800, 6.30691811549907882606758936220, 6.86816901578023874043324294053, 7.85982093784035209264751743846
