L(s) = 1 | − 2.58·2-s + 1.46·3-s + 4.69·4-s − 5-s − 3.80·6-s + 7-s − 6.98·8-s − 0.844·9-s + 2.58·10-s + 3.30·11-s + 6.89·12-s + 1.30·13-s − 2.58·14-s − 1.46·15-s + 8.68·16-s + 6.35·17-s + 2.18·18-s − 2.58·19-s − 4.69·20-s + 1.46·21-s − 8.56·22-s − 5.17·23-s − 10.2·24-s + 25-s − 3.38·26-s − 5.64·27-s + 4.69·28-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 0.847·3-s + 2.34·4-s − 0.447·5-s − 1.55·6-s + 0.377·7-s − 2.46·8-s − 0.281·9-s + 0.818·10-s + 0.997·11-s + 1.99·12-s + 0.363·13-s − 0.691·14-s − 0.379·15-s + 2.17·16-s + 1.54·17-s + 0.515·18-s − 0.591·19-s − 1.05·20-s + 0.320·21-s − 1.82·22-s − 1.07·23-s − 2.09·24-s + 0.200·25-s − 0.664·26-s − 1.08·27-s + 0.887·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 - 1.46T + 3T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 - 6.35T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 + 5.17T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 - 6.54T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 + 9.04T + 43T^{2} \) |
| 47 | \( 1 + 2.12T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 + 7.35T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 7.75T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 6.30T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 9.19T + 97T^{2} \) |
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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
−7.80262917040089983425215391929, −7.25615069176191832584324471766, −6.32116333402104235934759332563, −5.84178683189170299351777915107, −4.48218387190423437410792693218, −3.49502178495072178560006467160, −2.93432667315129174431088923421, −1.85321776246029950949876319302, −1.27899758500321722124702445155, 0,
1.27899758500321722124702445155, 1.85321776246029950949876319302, 2.93432667315129174431088923421, 3.49502178495072178560006467160, 4.48218387190423437410792693218, 5.84178683189170299351777915107, 6.32116333402104235934759332563, 7.25615069176191832584324471766, 7.80262917040089983425215391929