| L(s) = 1 | + 2.37·2-s − 3-s + 3.63·4-s + 3.47·5-s − 2.37·6-s + 3.87·8-s + 9-s + 8.24·10-s + 1.00·11-s − 3.63·12-s + 3.69·13-s − 3.47·15-s + 1.93·16-s − 2.26·17-s + 2.37·18-s − 3.08·19-s + 12.6·20-s + 2.39·22-s + 2.97·23-s − 3.87·24-s + 7.07·25-s + 8.77·26-s − 27-s + 6.10·29-s − 8.24·30-s − 1.10·31-s − 3.15·32-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 0.577·3-s + 1.81·4-s + 1.55·5-s − 0.969·6-s + 1.37·8-s + 0.333·9-s + 2.60·10-s + 0.304·11-s − 1.04·12-s + 1.02·13-s − 0.897·15-s + 0.484·16-s − 0.549·17-s + 0.559·18-s − 0.707·19-s + 2.82·20-s + 0.510·22-s + 0.619·23-s − 0.791·24-s + 1.41·25-s + 1.72·26-s − 0.192·27-s + 1.13·29-s − 1.50·30-s − 0.197·31-s − 0.557·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.792105209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.792105209\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 - 3.47T + 5T^{2} \) |
| 11 | \( 1 - 1.00T + 11T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 23 | \( 1 - 2.97T + 23T^{2} \) |
| 29 | \( 1 - 6.10T + 29T^{2} \) |
| 31 | \( 1 + 1.10T + 31T^{2} \) |
| 37 | \( 1 - 7.80T + 37T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 3.15T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.86T + 61T^{2} \) |
| 67 | \( 1 + 7.54T + 67T^{2} \) |
| 71 | \( 1 - 7.56T + 71T^{2} \) |
| 73 | \( 1 - 8.49T + 73T^{2} \) |
| 79 | \( 1 - 8.24T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 + 1.37T + 97T^{2} \) |
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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
−7.83296401263847885310902311085, −6.71815545949181507946422926929, −6.20659564488397028707730197202, −6.11931799164458775017348742978, −5.13815800261778791782317818576, −4.65402190428188787196580071332, −3.84002718063870328831134852586, −2.85515393312265068904819477270, −2.11764368047744518963061809073, −1.19814100491402109706838340286,
1.19814100491402109706838340286, 2.11764368047744518963061809073, 2.85515393312265068904819477270, 3.84002718063870328831134852586, 4.65402190428188787196580071332, 5.13815800261778791782317818576, 6.11931799164458775017348742978, 6.20659564488397028707730197202, 6.71815545949181507946422926929, 7.83296401263847885310902311085
