| L(s) = 1 | − 8·2-s − 71.1·3-s + 64·4-s + 523.·5-s + 569.·6-s − 416.·7-s − 512·8-s + 2.87e3·9-s − 4.18e3·10-s + 3.42e3·11-s − 4.55e3·12-s + 3.33e3·14-s − 3.72e4·15-s + 4.09e3·16-s − 6.53e3·17-s − 2.30e4·18-s + 2.79e4·19-s + 3.35e4·20-s + 2.96e4·21-s − 2.73e4·22-s + 1.06e5·23-s + 3.64e4·24-s + 1.95e5·25-s − 4.90e4·27-s − 2.66e4·28-s + 6.85e4·29-s + 2.98e5·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.52·3-s + 0.5·4-s + 1.87·5-s + 1.07·6-s − 0.459·7-s − 0.353·8-s + 1.31·9-s − 1.32·10-s + 0.775·11-s − 0.760·12-s + 0.324·14-s − 2.84·15-s + 0.250·16-s − 0.322·17-s − 0.930·18-s + 0.935·19-s + 0.936·20-s + 0.698·21-s − 0.548·22-s + 1.81·23-s + 0.537·24-s + 2.50·25-s − 0.479·27-s − 0.229·28-s + 0.521·29-s + 2.01·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.499747173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.499747173\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 71.1T + 2.18e3T^{2} \) |
| 5 | \( 1 - 523.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 416.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.42e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 6.53e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.79e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.06e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.85e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 5.15e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.81e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.02e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.16e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.26e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.34e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.17e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.89e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.18e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.28e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.67e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.09e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.87e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.80e6T + 8.07e13T^{2} \) |
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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
−10.31712903758051365512273099383, −9.563515161715840445579318589154, −8.913614366650806546256328320824, −7.01339613694661659856620335418, −6.46821681891209597350460649470, −5.70975673505344896551169212456, −4.88923701035266010361412830962, −2.87267500947223221718814603193, −1.47826896211398055483076364582, −0.77963620483203718354731703557,
0.77963620483203718354731703557, 1.47826896211398055483076364582, 2.87267500947223221718814603193, 4.88923701035266010361412830962, 5.70975673505344896551169212456, 6.46821681891209597350460649470, 7.01339613694661659856620335418, 8.913614366650806546256328320824, 9.563515161715840445579318589154, 10.31712903758051365512273099383
