| L(s) = 1 | + 1.34·2-s − 0.184·4-s − 0.879·5-s + 0.347·7-s − 2.94·8-s − 1.18·10-s + 2.22·11-s + 2.57·13-s + 0.467·14-s − 3.59·16-s − 0.467·17-s + 0.162·20-s + 3·22-s + 2.69·23-s − 4.22·25-s + 3.46·26-s − 0.0641·28-s + 6.87·29-s − 7.10·31-s + 1.04·32-s − 0.630·34-s − 0.305·35-s + 4.94·37-s + 2.58·40-s − 2.47·41-s + 3.90·43-s − 0.411·44-s + ⋯ |
| L(s) = 1 | + 0.952·2-s − 0.0923·4-s − 0.393·5-s + 0.131·7-s − 1.04·8-s − 0.374·10-s + 0.671·11-s + 0.713·13-s + 0.125·14-s − 0.899·16-s − 0.113·17-s + 0.0363·20-s + 0.639·22-s + 0.561·23-s − 0.845·25-s + 0.680·26-s − 0.0121·28-s + 1.27·29-s − 1.27·31-s + 0.184·32-s − 0.108·34-s − 0.0516·35-s + 0.812·37-s + 0.409·40-s − 0.386·41-s + 0.594·43-s − 0.0620·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.467175753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.467175753\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 + 0.879T + 5T^{2} \) |
| 7 | \( 1 - 0.347T + 7T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 - 2.57T + 13T^{2} \) |
| 17 | \( 1 + 0.467T + 17T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 29 | \( 1 - 6.87T + 29T^{2} \) |
| 31 | \( 1 + 7.10T + 31T^{2} \) |
| 37 | \( 1 - 4.94T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 - 3.90T + 43T^{2} \) |
| 47 | \( 1 - 7.29T + 47T^{2} \) |
| 53 | \( 1 - 2.83T + 53T^{2} \) |
| 59 | \( 1 - 6.30T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 7.67T + 67T^{2} \) |
| 71 | \( 1 - 9.30T + 71T^{2} \) |
| 73 | \( 1 - 1.38T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 9.45T + 97T^{2} \) |
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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
−8.681817981436187185655743159320, −7.943309167632514326205420498127, −6.94553176179274972945452465775, −6.25480272013218359017866379329, −5.52090850015382976653026956104, −4.71361440644148984373383496873, −3.92807268303585156764889591122, −3.44747244444419621416832950940, −2.28601552996149013103938692287, −0.829099153895638005552592750467,
0.829099153895638005552592750467, 2.28601552996149013103938692287, 3.44747244444419621416832950940, 3.92807268303585156764889591122, 4.71361440644148984373383496873, 5.52090850015382976653026956104, 6.25480272013218359017866379329, 6.94553176179274972945452465775, 7.943309167632514326205420498127, 8.681817981436187185655743159320
