| L(s) = 1 | − 3.02·2-s − 118.·4-s + 1.50e3·7-s + 746.·8-s − 1.59e3·11-s + 956.·13-s − 4.54e3·14-s + 1.29e4·16-s + 3.24e4·17-s − 3.91e4·19-s + 4.82e3·22-s + 5.93e4·23-s − 2.88e3·26-s − 1.78e5·28-s − 6.61e4·29-s − 1.96e4·31-s − 1.34e5·32-s − 9.81e4·34-s − 3.76e5·37-s + 1.18e5·38-s − 3.85e5·41-s − 4.66e5·43-s + 1.89e5·44-s − 1.79e5·46-s + 4.68e5·47-s + 1.44e6·49-s − 1.13e5·52-s + ⋯ |
| L(s) = 1 | − 0.267·2-s − 0.928·4-s + 1.65·7-s + 0.515·8-s − 0.361·11-s + 0.120·13-s − 0.443·14-s + 0.790·16-s + 1.60·17-s − 1.31·19-s + 0.0966·22-s + 1.01·23-s − 0.0322·26-s − 1.54·28-s − 0.503·29-s − 0.118·31-s − 0.726·32-s − 0.428·34-s − 1.22·37-s + 0.349·38-s − 0.872·41-s − 0.894·43-s + 0.335·44-s − 0.271·46-s + 0.658·47-s + 1.75·49-s − 0.112·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.796918537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.796918537\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3.02T + 128T^{2} \) |
| 7 | \( 1 - 1.50e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.59e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 956.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.24e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.91e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.93e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 6.61e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.96e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.76e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.85e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.66e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.68e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.60e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.04e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.78e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.64e3T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.79e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.01e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.76e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.06e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.14e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.02e6T + 8.07e13T^{2} \) |
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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
−10.81473789520593963443484959531, −10.09349334737264863945019602487, −8.760441283744589062578656677548, −8.237279596900896950230781282772, −7.28891112679951055835057577237, −5.48412871747952825319401322844, −4.85043072777476529358413051417, −3.66416911063841326671732278937, −1.85934588308764651544008600585, −0.77101206912823279184771740547,
0.77101206912823279184771740547, 1.85934588308764651544008600585, 3.66416911063841326671732278937, 4.85043072777476529358413051417, 5.48412871747952825319401322844, 7.28891112679951055835057577237, 8.237279596900896950230781282772, 8.760441283744589062578656677548, 10.09349334737264863945019602487, 10.81473789520593963443484959531
