| L(s) = 1 | − 3-s + 0.902·5-s − 3.04·7-s + 9-s − 2.68·11-s + 2.21·13-s − 0.902·15-s + 0.641·17-s + 2.31·19-s + 3.04·21-s − 3.98·23-s − 4.18·25-s − 27-s + 8.33·29-s + 4.78·31-s + 2.68·33-s − 2.74·35-s + 6.57·37-s − 2.21·39-s − 2.31·41-s − 8.29·43-s + 0.902·45-s + 3.72·47-s + 2.26·49-s − 0.641·51-s − 6.88·53-s − 2.42·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.403·5-s − 1.15·7-s + 0.333·9-s − 0.809·11-s + 0.615·13-s − 0.232·15-s + 0.155·17-s + 0.530·19-s + 0.664·21-s − 0.830·23-s − 0.837·25-s − 0.192·27-s + 1.54·29-s + 0.858·31-s + 0.467·33-s − 0.464·35-s + 1.08·37-s − 0.355·39-s − 0.361·41-s − 1.26·43-s + 0.134·45-s + 0.543·47-s + 0.323·49-s − 0.0897·51-s − 0.945·53-s − 0.326·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 - 0.902T + 5T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 - 2.21T + 13T^{2} \) |
| 17 | \( 1 - 0.641T + 17T^{2} \) |
| 19 | \( 1 - 2.31T + 19T^{2} \) |
| 23 | \( 1 + 3.98T + 23T^{2} \) |
| 29 | \( 1 - 8.33T + 29T^{2} \) |
| 31 | \( 1 - 4.78T + 31T^{2} \) |
| 37 | \( 1 - 6.57T + 37T^{2} \) |
| 41 | \( 1 + 2.31T + 41T^{2} \) |
| 43 | \( 1 + 8.29T + 43T^{2} \) |
| 47 | \( 1 - 3.72T + 47T^{2} \) |
| 53 | \( 1 + 6.88T + 53T^{2} \) |
| 59 | \( 1 - 1.34T + 59T^{2} \) |
| 61 | \( 1 - 2.11T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 9.93T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 4.87T + 89T^{2} \) |
| 97 | \( 1 - 6.66T + 97T^{2} \) |
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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
−7.80397489933475526414076550304, −6.71096802810535617012036140572, −6.34258295320996746964981254814, −5.69296325233959217658234580514, −4.96843727069354923663199564898, −4.04577957369984789938947970055, −3.18592638258830467230421386821, −2.41309860059292827271574783237, −1.16718502612057829036327612238, 0,
1.16718502612057829036327612238, 2.41309860059292827271574783237, 3.18592638258830467230421386821, 4.04577957369984789938947970055, 4.96843727069354923663199564898, 5.69296325233959217658234580514, 6.34258295320996746964981254814, 6.71096802810535617012036140572, 7.80397489933475526414076550304
