| L(s) = 1 | − 2.83e3·3-s + 1.05e5·5-s − 5.01e5·7-s − 6.29e6·9-s − 9.46e7·11-s + 2.61e8·13-s − 2.98e8·15-s + 2.42e9·17-s − 1.15e9·19-s + 1.42e9·21-s + 2.23e10·23-s − 1.94e10·25-s + 5.85e10·27-s + 1.12e11·29-s + 1.67e11·31-s + 2.68e11·33-s − 5.28e10·35-s + 7.53e11·37-s − 7.41e11·39-s − 1.42e12·41-s + 1.58e12·43-s − 6.63e11·45-s + 1.13e12·47-s − 4.49e12·49-s − 6.88e12·51-s − 5.07e12·53-s − 9.97e12·55-s + ⋯ |
| L(s) = 1 | − 0.749·3-s + 0.602·5-s − 0.230·7-s − 0.438·9-s − 1.46·11-s + 1.15·13-s − 0.451·15-s + 1.43·17-s − 0.296·19-s + 0.172·21-s + 1.37·23-s − 0.636·25-s + 1.07·27-s + 1.21·29-s + 1.09·31-s + 1.09·33-s − 0.138·35-s + 1.30·37-s − 0.865·39-s − 1.14·41-s + 0.889·43-s − 0.264·45-s + 0.325·47-s − 0.946·49-s − 1.07·51-s − 0.593·53-s − 0.883·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.443493089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.443493089\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 2.83e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 1.05e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 5.01e5T + 4.74e12T^{2} \) |
| 11 | \( 1 + 9.46e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 2.61e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 2.42e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 1.15e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.23e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 1.12e11T + 8.62e21T^{2} \) |
| 31 | \( 1 - 1.67e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 7.53e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 1.42e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 1.58e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 1.13e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 5.07e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 1.15e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 1.56e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 7.82e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 6.47e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 7.33e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 1.87e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 4.39e13T + 6.11e28T^{2} \) |
| 89 | \( 1 + 5.90e13T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.06e15T + 6.33e29T^{2} \) |
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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
−15.64436018072153669198806343263, −13.91783694385469392075864340310, −12.71327040351165631588581608449, −11.17297734284009146202490202446, −10.04083493988207157462660641592, −8.210374870027580357505834850846, −6.23950975694699971103197654089, −5.19723294813748792731029957924, −2.90079594633906577563969189815, −0.846061202646158350681722241755,
0.846061202646158350681722241755, 2.90079594633906577563969189815, 5.19723294813748792731029957924, 6.23950975694699971103197654089, 8.210374870027580357505834850846, 10.04083493988207157462660641592, 11.17297734284009146202490202446, 12.71327040351165631588581608449, 13.91783694385469392075864340310, 15.64436018072153669198806343263
