| 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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 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 | \( 1 - 9.70e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.662832269246255572626987547961, −8.838171717235767958524798252017, −7.65396922092859854234575962048, −6.79048829064341547232775642947, −5.68148591647934045465838994637, −5.05423408088352400972026811916, −3.66839276282080051094397199809, −2.56682609373032899327127390400, −1.08793158673093796766060499531, 0,
1.08793158673093796766060499531, 2.56682609373032899327127390400, 3.66839276282080051094397199809, 5.05423408088352400972026811916, 5.68148591647934045465838994637, 6.79048829064341547232775642947, 7.65396922092859854234575962048, 8.838171717235767958524798252017, 9.662832269246255572626987547961
