| L(s) = 1 | + 2.60·3-s − 1.83·7-s + 3.77·9-s − 2.57·11-s − 1.16·13-s − 4.37·17-s + 8.21·19-s − 4.77·21-s − 5.87·23-s + 2.02·27-s + 2.45·29-s + 1.96·31-s − 6.70·33-s + 37-s − 3.04·39-s − 4.40·41-s − 10.9·43-s − 2.19·47-s − 3.63·49-s − 11.3·51-s + 6.44·53-s + 21.3·57-s − 5.31·59-s − 1.92·61-s − 6.93·63-s − 9.58·67-s − 15.2·69-s + ⋯ |
| L(s) = 1 | + 1.50·3-s − 0.693·7-s + 1.25·9-s − 0.776·11-s − 0.324·13-s − 1.06·17-s + 1.88·19-s − 1.04·21-s − 1.22·23-s + 0.390·27-s + 0.455·29-s + 0.353·31-s − 1.16·33-s + 0.164·37-s − 0.487·39-s − 0.688·41-s − 1.66·43-s − 0.319·47-s − 0.519·49-s − 1.59·51-s + 0.884·53-s + 2.83·57-s − 0.691·59-s − 0.245·61-s − 0.873·63-s − 1.17·67-s − 1.84·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 11 | \( 1 + 2.57T + 11T^{2} \) |
| 13 | \( 1 + 1.16T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 8.21T + 19T^{2} \) |
| 23 | \( 1 + 5.87T + 23T^{2} \) |
| 29 | \( 1 - 2.45T + 29T^{2} \) |
| 31 | \( 1 - 1.96T + 31T^{2} \) |
| 41 | \( 1 + 4.40T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 5.31T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 + 9.58T + 67T^{2} \) |
| 71 | \( 1 - 3.34T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 7.84T + 79T^{2} \) |
| 83 | \( 1 - 5.94T + 83T^{2} \) |
| 89 | \( 1 + 4.71T + 89T^{2} \) |
| 97 | \( 1 - 4.14T + 97T^{2} \) |
| 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
−7.77380438221412235634322403413, −7.02392613848248657859546204110, −6.32968153020519332969779350118, −5.35354250813643235740755872156, −4.59087320496417846686539323993, −3.66248235735476381116198262435, −3.06085589887160388742016133900, −2.49253999766850301263740328006, −1.57425735451752144925734281963, 0,
1.57425735451752144925734281963, 2.49253999766850301263740328006, 3.06085589887160388742016133900, 3.66248235735476381116198262435, 4.59087320496417846686539323993, 5.35354250813643235740755872156, 6.32968153020519332969779350118, 7.02392613848248657859546204110, 7.77380438221412235634322403413
