| L(s) = 1 | − 42.3·3-s − 290.·5-s − 951.·7-s − 393.·9-s + 1.38e3·11-s − 1.44e3·13-s + 1.23e4·15-s + 1.74e4·17-s + 6.85e3·19-s + 4.02e4·21-s + 3.83e4·23-s + 6.53e3·25-s + 1.09e5·27-s − 1.82e4·29-s + 1.41e5·31-s − 5.87e4·33-s + 2.76e5·35-s − 7.79e4·37-s + 6.13e4·39-s − 1.81e5·41-s + 3.62e5·43-s + 1.14e5·45-s + 2.80e5·47-s + 8.18e4·49-s − 7.38e5·51-s + 7.00e5·53-s − 4.03e5·55-s + ⋯ |
| L(s) = 1 | − 0.905·3-s − 1.04·5-s − 1.04·7-s − 0.179·9-s + 0.314·11-s − 0.183·13-s + 0.942·15-s + 0.860·17-s + 0.229·19-s + 0.949·21-s + 0.656·23-s + 0.0836·25-s + 1.06·27-s − 0.139·29-s + 0.854·31-s − 0.284·33-s + 1.09·35-s − 0.253·37-s + 0.165·39-s − 0.410·41-s + 0.695·43-s + 0.187·45-s + 0.394·47-s + 0.0993·49-s − 0.779·51-s + 0.646·53-s − 0.327·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 3 | \( 1 + 42.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 290.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 951.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.38e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.44e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.74e4T + 4.10e8T^{2} \) |
| 23 | \( 1 - 3.83e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.82e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.41e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 7.79e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.81e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.62e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.80e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.00e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.01e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.06e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 6.78e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.12e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.01e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.03e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.58e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.37e6T + 8.07e13T^{2} \) |
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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
−10.17341072854689058494115182098, −9.159333072895536959371536283906, −8.011119190250785783484210579552, −6.99884568541504576199662877069, −6.12297930455095538422941729325, −5.10359482021059917501003114085, −3.86039663799314268037173337576, −2.91124630338979037024800679440, −0.900592079730861912852551054748, 0,
0.900592079730861912852551054748, 2.91124630338979037024800679440, 3.86039663799314268037173337576, 5.10359482021059917501003114085, 6.12297930455095538422941729325, 6.99884568541504576199662877069, 8.011119190250785783484210579552, 9.159333072895536959371536283906, 10.17341072854689058494115182098
