L(s) = 1 | − 0.486·3-s − 3.69·5-s − 0.0825·7-s − 2.76·9-s − 5.07·11-s + 2.31·13-s + 1.79·15-s − 8.07·17-s + 4.31·19-s + 0.0401·21-s + 7.96·23-s + 8.68·25-s + 2.80·27-s + 1.34·29-s + 1.41·31-s + 2.46·33-s + 0.305·35-s + 5.54·37-s − 1.12·39-s + 7.34·41-s + 5.81·43-s + 10.2·45-s − 7.86·47-s − 6.99·49-s + 3.92·51-s + 0.710·53-s + 18.7·55-s + ⋯ |
L(s) = 1 | − 0.280·3-s − 1.65·5-s − 0.0311·7-s − 0.921·9-s − 1.53·11-s + 0.641·13-s + 0.464·15-s − 1.95·17-s + 0.989·19-s + 0.00875·21-s + 1.66·23-s + 1.73·25-s + 0.539·27-s + 0.250·29-s + 0.253·31-s + 0.429·33-s + 0.0516·35-s + 0.911·37-s − 0.180·39-s + 1.14·41-s + 0.886·43-s + 1.52·45-s − 1.14·47-s − 0.999·49-s + 0.549·51-s + 0.0975·53-s + 2.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 + 0.486T + 3T^{2} \) |
| 5 | \( 1 + 3.69T + 5T^{2} \) |
| 7 | \( 1 + 0.0825T + 7T^{2} \) |
| 11 | \( 1 + 5.07T + 11T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 + 8.07T + 17T^{2} \) |
| 19 | \( 1 - 4.31T + 19T^{2} \) |
| 23 | \( 1 - 7.96T + 23T^{2} \) |
| 29 | \( 1 - 1.34T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 - 5.54T + 37T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 - 5.81T + 43T^{2} \) |
| 47 | \( 1 + 7.86T + 47T^{2} \) |
| 53 | \( 1 - 0.710T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 - 9.64T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 8.05T + 73T^{2} \) |
| 79 | \( 1 - 0.177T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 - 2.68T + 89T^{2} \) |
| 97 | \( 1 + 5.91T + 97T^{2} \) |
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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
−7.52662166173431985589863476314, −6.93390539662229289593109491700, −6.13858154273678931186583100548, −5.21343096782516816480481860390, −4.72378437383623267379218796370, −3.94746882316619959454566706189, −2.99782226404736761041303357279, −2.59650455380953603885156247541, −0.855614144301243185271741494955, 0,
0.855614144301243185271741494955, 2.59650455380953603885156247541, 2.99782226404736761041303357279, 3.94746882316619959454566706189, 4.72378437383623267379218796370, 5.21343096782516816480481860390, 6.13858154273678931186583100548, 6.93390539662229289593109491700, 7.52662166173431985589863476314