| L(s) = 1 | + 1.53·3-s + 2.22·7-s − 0.652·9-s − 3.22·11-s − 1.57·13-s − 3.53·17-s + 19-s + 3.41·21-s − 4.47·23-s − 5.59·27-s + 1.92·29-s − 3.81·31-s − 4.94·33-s − 11.3·37-s − 2.41·39-s + 3.47·41-s + 1.69·43-s − 1.57·47-s − 2.04·49-s − 5.41·51-s + 7.12·53-s + 1.53·57-s − 7.88·59-s − 2.79·61-s − 1.45·63-s − 3.22·67-s − 6.85·69-s + ⋯ |
| L(s) = 1 | + 0.884·3-s + 0.841·7-s − 0.217·9-s − 0.972·11-s − 0.436·13-s − 0.856·17-s + 0.229·19-s + 0.744·21-s − 0.933·23-s − 1.07·27-s + 0.356·29-s − 0.685·31-s − 0.860·33-s − 1.86·37-s − 0.386·39-s + 0.542·41-s + 0.258·43-s − 0.229·47-s − 0.291·49-s − 0.757·51-s + 0.979·53-s + 0.202·57-s − 1.02·59-s − 0.357·61-s − 0.183·63-s − 0.394·67-s − 0.825·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 7 | \( 1 - 2.22T + 7T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 + 3.53T + 17T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 3.47T + 41T^{2} \) |
| 43 | \( 1 - 1.69T + 43T^{2} \) |
| 47 | \( 1 + 1.57T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 + 7.88T + 59T^{2} \) |
| 61 | \( 1 + 2.79T + 61T^{2} \) |
| 67 | \( 1 + 3.22T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 - 6.41T + 73T^{2} \) |
| 79 | \( 1 + 8.59T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 6.10T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.075090097932257964949754669774, −7.68709264636382567784982554121, −6.83811836414562978227246153356, −5.75172833104877227755021403517, −5.09305823951035901966876970411, −4.27372445316006002731419443181, −3.31085828372656333225823796987, −2.44176648847454841233427057372, −1.78364103203150793156084765108, 0,
1.78364103203150793156084765108, 2.44176648847454841233427057372, 3.31085828372656333225823796987, 4.27372445316006002731419443181, 5.09305823951035901966876970411, 5.75172833104877227755021403517, 6.83811836414562978227246153356, 7.68709264636382567784982554121, 8.075090097932257964949754669774
