| L(s) = 1 | − 1.90·2-s − 2.27·3-s + 1.64·4-s + 5-s + 4.34·6-s − 0.428·7-s + 0.683·8-s + 2.19·9-s − 1.90·10-s + 4.71·11-s − 3.74·12-s + 3.29·13-s + 0.818·14-s − 2.27·15-s − 4.58·16-s + 17-s − 4.18·18-s − 3.08·19-s + 1.64·20-s + 0.977·21-s − 8.99·22-s + 4.29·23-s − 1.55·24-s + 25-s − 6.28·26-s + 1.83·27-s − 0.704·28-s + ⋯ |
| L(s) = 1 | − 1.34·2-s − 1.31·3-s + 0.820·4-s + 0.447·5-s + 1.77·6-s − 0.162·7-s + 0.241·8-s + 0.731·9-s − 0.603·10-s + 1.42·11-s − 1.08·12-s + 0.912·13-s + 0.218·14-s − 0.588·15-s − 1.14·16-s + 0.242·17-s − 0.986·18-s − 0.708·19-s + 0.367·20-s + 0.213·21-s − 1.91·22-s + 0.895·23-s − 0.317·24-s + 0.200·25-s − 1.23·26-s + 0.353·27-s − 0.133·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 + 1.90T + 2T^{2} \) |
| 3 | \( 1 + 2.27T + 3T^{2} \) |
| 7 | \( 1 + 0.428T + 7T^{2} \) |
| 11 | \( 1 - 4.71T + 11T^{2} \) |
| 13 | \( 1 - 3.29T + 13T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 + 6.65T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 - 4.22T + 37T^{2} \) |
| 41 | \( 1 - 0.578T + 41T^{2} \) |
| 43 | \( 1 + 5.00T + 43T^{2} \) |
| 47 | \( 1 + 3.57T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 4.06T + 67T^{2} \) |
| 73 | \( 1 - 6.53T + 73T^{2} \) |
| 79 | \( 1 + 6.52T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 - 5.99T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−7.84747796227649884326407288247, −6.82503487632417951615189659904, −6.43125389310439630962181606369, −5.94643718429640692638155547019, −4.88966710169741798885764089358, −4.24882779646908436898361964225, −3.09064500348405637439584436770, −1.61848078459107599160658082530, −1.15173142357853470766972146638, 0,
1.15173142357853470766972146638, 1.61848078459107599160658082530, 3.09064500348405637439584436770, 4.24882779646908436898361964225, 4.88966710169741798885764089358, 5.94643718429640692638155547019, 6.43125389310439630962181606369, 6.82503487632417951615189659904, 7.84747796227649884326407288247
