| L(s) = 1 | − 0.732·3-s − 2.73·7-s − 2.46·9-s − 2·11-s + 3.46·13-s + 3.46·17-s − 7.46·19-s + 2·21-s + 4.19·23-s + 4·27-s + 6.92·29-s + 1.46·31-s + 1.46·33-s + 2·37-s − 2.53·39-s + 5.46·41-s + 8.73·43-s − 6.73·47-s + 0.464·49-s − 2.53·51-s − 4.53·53-s + 5.46·57-s − 0.535·59-s − 4.92·61-s + 6.73·63-s + 7.26·67-s − 3.07·69-s + ⋯ |
| L(s) = 1 | − 0.422·3-s − 1.03·7-s − 0.821·9-s − 0.603·11-s + 0.960·13-s + 0.840·17-s − 1.71·19-s + 0.436·21-s + 0.874·23-s + 0.769·27-s + 1.28·29-s + 0.262·31-s + 0.254·33-s + 0.328·37-s − 0.406·39-s + 0.853·41-s + 1.33·43-s − 0.981·47-s + 0.0663·49-s − 0.355·51-s − 0.623·53-s + 0.723·57-s − 0.0697·59-s − 0.630·61-s + 0.848·63-s + 0.887·67-s − 0.369·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 5.46T + 41T^{2} \) |
| 43 | \( 1 - 8.73T + 43T^{2} \) |
| 47 | \( 1 + 6.73T + 47T^{2} \) |
| 53 | \( 1 + 4.53T + 53T^{2} \) |
| 59 | \( 1 + 0.535T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 7.26T + 67T^{2} \) |
| 71 | \( 1 - 1.46T + 71T^{2} \) |
| 73 | \( 1 - 0.535T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
| 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
−7.77237609638573942369887945976, −6.65559154866201543404930637388, −6.30328675373665937973372476588, −5.70335793892666390632137277159, −4.86955597835043821188753150976, −3.99317622618812923927802151972, −3.07604351301349161789284465520, −2.55605404189965930725500557986, −1.08054165479826905556508312822, 0,
1.08054165479826905556508312822, 2.55605404189965930725500557986, 3.07604351301349161789284465520, 3.99317622618812923927802151972, 4.86955597835043821188753150976, 5.70335793892666390632137277159, 6.30328675373665937973372476588, 6.65559154866201543404930637388, 7.77237609638573942369887945976
