| L(s) = 1 | + 30.2·3-s + 161.·7-s − 1.27e3·9-s + 1.11e3·11-s + 5.15e3·13-s − 7.12e3·17-s + 3.39e4·19-s + 4.87e3·21-s − 2.44e4·23-s − 1.04e5·27-s − 1.60e4·29-s − 1.14e5·31-s + 3.38e4·33-s + 4.92e5·37-s + 1.55e5·39-s + 6.40e5·41-s − 6.93e4·43-s − 1.21e6·47-s − 7.97e5·49-s − 2.15e5·51-s + 1.38e6·53-s + 1.02e6·57-s + 2.49e6·59-s + 1.83e6·61-s − 2.05e5·63-s − 1.94e6·67-s − 7.40e5·69-s + ⋯ |
| L(s) = 1 | + 0.645·3-s + 0.177·7-s − 0.582·9-s + 0.253·11-s + 0.650·13-s − 0.351·17-s + 1.13·19-s + 0.114·21-s − 0.419·23-s − 1.02·27-s − 0.122·29-s − 0.692·31-s + 0.163·33-s + 1.59·37-s + 0.420·39-s + 1.45·41-s − 0.132·43-s − 1.71·47-s − 0.968·49-s − 0.227·51-s + 1.28·53-s + 0.733·57-s + 1.58·59-s + 1.03·61-s − 0.103·63-s − 0.788·67-s − 0.271·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.833504149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.833504149\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 30.2T + 2.18e3T^{2} \) |
| 7 | \( 1 - 161.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.11e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.15e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 7.12e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.39e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.44e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.60e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.14e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.92e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.40e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.93e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.21e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.38e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.49e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.83e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.94e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.63e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.95e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.31e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.04e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.83e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.70e6T + 8.07e13T^{2} \) |
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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
−9.891913758078800956876015481280, −9.130404232983271572035713022492, −8.293829240346230925599470448471, −7.51616559285694248795215285244, −6.29421217747817902709576569058, −5.36074498393359574108176383258, −4.04706344501289454552570966313, −3.11404695243624213309227235565, −2.03040550949409252301936040585, −0.75243487717403128946143953758,
0.75243487717403128946143953758, 2.03040550949409252301936040585, 3.11404695243624213309227235565, 4.04706344501289454552570966313, 5.36074498393359574108176383258, 6.29421217747817902709576569058, 7.51616559285694248795215285244, 8.293829240346230925599470448471, 9.130404232983271572035713022492, 9.891913758078800956876015481280
