| L(s) = 1 | + 2.68·2-s + 3-s + 5.22·4-s + 3.75·5-s + 2.68·6-s − 7-s + 8.65·8-s + 9-s + 10.0·10-s − 2.87·11-s + 5.22·12-s + 3.73·13-s − 2.68·14-s + 3.75·15-s + 12.8·16-s + 5.21·17-s + 2.68·18-s − 4.70·19-s + 19.6·20-s − 21-s − 7.72·22-s + 2.09·23-s + 8.65·24-s + 9.11·25-s + 10.0·26-s + 27-s − 5.22·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 0.577·3-s + 2.61·4-s + 1.67·5-s + 1.09·6-s − 0.377·7-s + 3.06·8-s + 0.333·9-s + 3.19·10-s − 0.866·11-s + 1.50·12-s + 1.03·13-s − 0.718·14-s + 0.969·15-s + 3.20·16-s + 1.26·17-s + 0.633·18-s − 1.07·19-s + 4.38·20-s − 0.218·21-s − 1.64·22-s + 0.436·23-s + 1.76·24-s + 1.82·25-s + 1.96·26-s + 0.192·27-s − 0.986·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.53171280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.53171280\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 5 | \( 1 - 3.75T + 5T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 - 5.21T + 17T^{2} \) |
| 19 | \( 1 + 4.70T + 19T^{2} \) |
| 23 | \( 1 - 2.09T + 23T^{2} \) |
| 29 | \( 1 + 1.32T + 29T^{2} \) |
| 31 | \( 1 + 9.51T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 + 3.48T + 41T^{2} \) |
| 43 | \( 1 + 0.719T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 - 0.00850T + 53T^{2} \) |
| 59 | \( 1 - 0.526T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 9.27T + 73T^{2} \) |
| 79 | \( 1 - 0.689T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 1.31T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−7.48391607035927158732834697047, −6.83579492188178607339530792379, −6.11742263703931550370191589200, −5.68313818503127887024341597000, −5.17615667295315150662571559342, −4.31819286618109477416866888872, −3.27594231801308169565772312668, −3.06501067727413072165010458990, −1.96789591095718616450608309210, −1.60219583594470649347499596764,
1.60219583594470649347499596764, 1.96789591095718616450608309210, 3.06501067727413072165010458990, 3.27594231801308169565772312668, 4.31819286618109477416866888872, 5.17615667295315150662571559342, 5.68313818503127887024341597000, 6.11742263703931550370191589200, 6.83579492188178607339530792379, 7.48391607035927158732834697047
