| L(s) = 1 | − 2.73·3-s + 0.732·7-s + 4.46·9-s − 1.26·11-s + 2·17-s − 1.26·19-s − 2·21-s + 1.46·23-s − 5·25-s − 3.99·27-s − 6.92·29-s − 1.46·31-s + 3.46·33-s − 10.9·37-s − 41-s − 6.92·43-s − 6.19·47-s − 6.46·49-s − 5.46·51-s + 6.92·53-s + 3.46·57-s − 9.46·59-s − 10·61-s + 3.26·63-s + 11.1·67-s − 4·69-s + 7.26·71-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 0.276·7-s + 1.48·9-s − 0.382·11-s + 0.485·17-s − 0.290·19-s − 0.436·21-s + 0.305·23-s − 25-s − 0.769·27-s − 1.28·29-s − 0.262·31-s + 0.603·33-s − 1.79·37-s − 0.156·41-s − 1.05·43-s − 0.903·47-s − 0.923·49-s − 0.765·51-s + 0.951·53-s + 0.458·57-s − 1.23·59-s − 1.28·61-s + 0.411·63-s + 1.35·67-s − 0.481·69-s + 0.862·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 7.26T + 71T^{2} \) |
| 73 | \( 1 - 8.39T + 73T^{2} \) |
| 79 | \( 1 + 7.66T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−10.34725221754258365402833607973, −9.498928604503850384170155359842, −8.238247403149895657797078121315, −7.27197083446314853338412195162, −6.38268733816239713774387999574, −5.47984513367997500475954256242, −4.89276539195136279814845963132, −3.60064913139710519609432106987, −1.71220995929975346120121673799, 0,
1.71220995929975346120121673799, 3.60064913139710519609432106987, 4.89276539195136279814845963132, 5.47984513367997500475954256242, 6.38268733816239713774387999574, 7.27197083446314853338412195162, 8.238247403149895657797078121315, 9.498928604503850384170155359842, 10.34725221754258365402833607973
