| L(s) = 1 | − 86.9·3-s − 188.·5-s − 639.·7-s + 5.37e3·9-s + 1.62e3·11-s + 1.13e3·13-s + 1.63e4·15-s + 2.87e4·17-s − 4.33e4·19-s + 5.55e4·21-s + 1.21e4·23-s − 4.26e4·25-s − 2.77e5·27-s + 6.25e4·29-s − 2.25e4·31-s − 1.41e5·33-s + 1.20e5·35-s − 2.77e5·37-s − 9.89e4·39-s − 1.30e5·41-s − 5.85e5·43-s − 1.01e6·45-s + 9.93e4·47-s − 4.15e5·49-s − 2.49e6·51-s − 6.36e5·53-s − 3.06e5·55-s + ⋯ |
| L(s) = 1 | − 1.85·3-s − 0.673·5-s − 0.704·7-s + 2.45·9-s + 0.369·11-s + 0.143·13-s + 1.25·15-s + 1.41·17-s − 1.45·19-s + 1.30·21-s + 0.208·23-s − 0.546·25-s − 2.70·27-s + 0.476·29-s − 0.135·31-s − 0.686·33-s + 0.474·35-s − 0.900·37-s − 0.267·39-s − 0.294·41-s − 1.12·43-s − 1.65·45-s + 0.139·47-s − 0.503·49-s − 2.63·51-s − 0.587·53-s − 0.248·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3848827268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3848827268\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 3 | \( 1 + 86.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 188.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 639.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.62e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.13e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.87e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.33e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 6.25e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.25e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.30e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.85e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.93e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.36e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.08e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.40e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.05e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.78e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.20e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.98e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.57e6T + 8.07e13T^{2} \) |
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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
−10.40008320238059339420616893071, −9.666785718985313604333366229411, −8.206535568196105467353245238765, −7.04085625635044953835923162658, −6.36251799727899310867471302381, −5.50794051219726010572274741530, −4.45927023879886829267660600631, −3.49806086376224620548977375035, −1.49274865737193555948035871879, −0.33707820438600403481576983133,
0.33707820438600403481576983133, 1.49274865737193555948035871879, 3.49806086376224620548977375035, 4.45927023879886829267660600631, 5.50794051219726010572274741530, 6.36251799727899310867471302381, 7.04085625635044953835923162658, 8.206535568196105467353245238765, 9.666785718985313604333366229411, 10.40008320238059339420616893071
