L(s) = 1 | − 2.20·3-s + 3.30·5-s − 4.05·7-s + 1.86·9-s − 2.06·11-s − 0.105·13-s − 7.28·15-s + 17-s − 3.19·19-s + 8.94·21-s + 1.62·23-s + 5.90·25-s + 2.51·27-s − 0.938·29-s + 0.822·31-s + 4.55·33-s − 13.3·35-s + 7.92·37-s + 0.232·39-s + 1.99·41-s − 9.90·43-s + 6.14·45-s + 7.98·47-s + 9.45·49-s − 2.20·51-s + 3.19·53-s − 6.82·55-s + ⋯ |
L(s) = 1 | − 1.27·3-s + 1.47·5-s − 1.53·7-s + 0.620·9-s − 0.622·11-s − 0.0292·13-s − 1.88·15-s + 0.242·17-s − 0.732·19-s + 1.95·21-s + 0.339·23-s + 1.18·25-s + 0.483·27-s − 0.174·29-s + 0.147·31-s + 0.792·33-s − 2.26·35-s + 1.30·37-s + 0.0371·39-s + 0.311·41-s − 1.51·43-s + 0.916·45-s + 1.16·47-s + 1.35·49-s − 0.308·51-s + 0.439·53-s − 0.919·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 2.20T + 3T^{2} \) |
| 5 | \( 1 - 3.30T + 5T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 11 | \( 1 + 2.06T + 11T^{2} \) |
| 13 | \( 1 + 0.105T + 13T^{2} \) |
| 19 | \( 1 + 3.19T + 19T^{2} \) |
| 23 | \( 1 - 1.62T + 23T^{2} \) |
| 29 | \( 1 + 0.938T + 29T^{2} \) |
| 31 | \( 1 - 0.822T + 31T^{2} \) |
| 37 | \( 1 - 7.92T + 37T^{2} \) |
| 41 | \( 1 - 1.99T + 41T^{2} \) |
| 43 | \( 1 + 9.90T + 43T^{2} \) |
| 47 | \( 1 - 7.98T + 47T^{2} \) |
| 53 | \( 1 - 3.19T + 53T^{2} \) |
| 61 | \( 1 - 5.05T + 61T^{2} \) |
| 67 | \( 1 - 7.19T + 67T^{2} \) |
| 71 | \( 1 - 9.80T + 71T^{2} \) |
| 73 | \( 1 + 9.52T + 73T^{2} \) |
| 79 | \( 1 - 9.13T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 1.21T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.06469719408758404206156018894, −6.57100635245769731165449039740, −6.05968854025559840368219208906, −5.58215623416853145728932377395, −5.01941064326505096792675164184, −3.99174274844095929693382974486, −2.90642691898384912862277938601, −2.30217230860897094846386741918, −1.04133966844219800214970696070, 0,
1.04133966844219800214970696070, 2.30217230860897094846386741918, 2.90642691898384912862277938601, 3.99174274844095929693382974486, 5.01941064326505096792675164184, 5.58215623416853145728932377395, 6.05968854025559840368219208906, 6.57100635245769731165449039740, 7.06469719408758404206156018894