| L(s) = 1 | − 1.46·3-s + 3.99·5-s − 0.850·9-s + 0.414·11-s − 6.60·13-s − 5.85·15-s − 17-s + 1.00·19-s − 7.41·23-s + 10.9·25-s + 5.64·27-s − 2.31·29-s + 4.19·31-s − 0.608·33-s + 11.9·37-s + 9.67·39-s + 4.57·41-s + 2.78·43-s − 3.39·45-s − 7.49·47-s + 1.46·51-s + 7.65·53-s + 1.65·55-s − 1.47·57-s − 3.29·59-s − 3.09·61-s − 26.3·65-s + ⋯ |
| L(s) = 1 | − 0.846·3-s + 1.78·5-s − 0.283·9-s + 0.125·11-s − 1.83·13-s − 1.51·15-s − 0.242·17-s + 0.230·19-s − 1.54·23-s + 2.19·25-s + 1.08·27-s − 0.429·29-s + 0.754·31-s − 0.105·33-s + 1.96·37-s + 1.54·39-s + 0.715·41-s + 0.424·43-s − 0.506·45-s − 1.09·47-s + 0.205·51-s + 1.05·53-s + 0.223·55-s − 0.194·57-s − 0.429·59-s − 0.396·61-s − 3.26·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.46T + 3T^{2} \) |
| 5 | \( 1 - 3.99T + 5T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 13 | \( 1 + 6.60T + 13T^{2} \) |
| 19 | \( 1 - 1.00T + 19T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 29 | \( 1 + 2.31T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 4.57T + 41T^{2} \) |
| 43 | \( 1 - 2.78T + 43T^{2} \) |
| 47 | \( 1 + 7.49T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 + 3.09T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 9.56T + 71T^{2} \) |
| 73 | \( 1 - 7.17T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 3.80T + 83T^{2} \) |
| 89 | \( 1 + 2.06T + 89T^{2} \) |
| 97 | \( 1 + 1.03T + 97T^{2} \) |
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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
−7.50427501323791225314210925663, −6.71754872629001755218522166736, −5.97074910708008875504686376564, −5.75920569172710678477249318038, −4.93026889742559943572985774468, −4.33943407423163768096638302222, −2.74505540690296328935330625576, −2.39165555752981310981417738096, −1.31970446111376064636666757036, 0,
1.31970446111376064636666757036, 2.39165555752981310981417738096, 2.74505540690296328935330625576, 4.33943407423163768096638302222, 4.93026889742559943572985774468, 5.75920569172710678477249318038, 5.97074910708008875504686376564, 6.71754872629001755218522166736, 7.50427501323791225314210925663
