| L(s) = 1 | + 1.83·2-s − 0.0182·3-s + 1.37·4-s + 1.49·5-s − 0.0334·6-s − 3.57·7-s − 1.14·8-s − 2.99·9-s + 2.74·10-s − 11-s − 0.0251·12-s − 1.26·13-s − 6.57·14-s − 0.0271·15-s − 4.85·16-s − 4.58·17-s − 5.51·18-s + 3.55·19-s + 2.05·20-s + 0.0652·21-s − 1.83·22-s − 3.11·23-s + 0.0208·24-s − 2.77·25-s − 2.33·26-s + 0.109·27-s − 4.92·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 0.0105·3-s + 0.688·4-s + 0.666·5-s − 0.0136·6-s − 1.35·7-s − 0.404·8-s − 0.999·9-s + 0.866·10-s − 0.301·11-s − 0.00724·12-s − 0.351·13-s − 1.75·14-s − 0.00701·15-s − 1.21·16-s − 1.11·17-s − 1.29·18-s + 0.814·19-s + 0.459·20-s + 0.0142·21-s − 0.391·22-s − 0.649·23-s + 0.00425·24-s − 0.555·25-s − 0.456·26-s + 0.0210·27-s − 0.931·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 3 | \( 1 + 0.0182T + 3T^{2} \) |
| 5 | \( 1 - 1.49T + 5T^{2} \) |
| 7 | \( 1 + 3.57T + 7T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 + 3.11T + 23T^{2} \) |
| 29 | \( 1 - 5.18T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 0.879T + 41T^{2} \) |
| 43 | \( 1 + 0.823T + 43T^{2} \) |
| 47 | \( 1 - 0.232T + 47T^{2} \) |
| 53 | \( 1 + 2.76T + 53T^{2} \) |
| 59 | \( 1 + 1.50T + 59T^{2} \) |
| 61 | \( 1 - 4.67T + 61T^{2} \) |
| 67 | \( 1 + 6.86T + 67T^{2} \) |
| 71 | \( 1 - 1.57T + 71T^{2} \) |
| 73 | \( 1 - 2.77T + 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 - 3.47T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−9.377866928386685824968912162688, −8.394330027756703485459683826538, −7.14566668115918896298694918351, −6.19422200228644822494508824603, −5.90358704649236170151559414652, −4.99408151500542715048990721008, −3.96499320697485214170885936731, −2.99648538556932982315303619239, −2.39468281638331517032225096615, 0,
2.39468281638331517032225096615, 2.99648538556932982315303619239, 3.96499320697485214170885936731, 4.99408151500542715048990721008, 5.90358704649236170151559414652, 6.19422200228644822494508824603, 7.14566668115918896298694918351, 8.394330027756703485459683826538, 9.377866928386685824968912162688
