| L(s) = 1 | + 0.298·2-s − 3-s − 1.91·4-s + 2.54·5-s − 0.298·6-s − 7-s − 1.16·8-s + 9-s + 0.759·10-s + 6.15·11-s + 1.91·12-s + 4.68·13-s − 0.298·14-s − 2.54·15-s + 3.47·16-s − 4.41·17-s + 0.298·18-s − 5.03·19-s − 4.86·20-s + 21-s + 1.83·22-s + 9.11·23-s + 1.16·24-s + 1.48·25-s + 1.39·26-s − 27-s + 1.91·28-s + ⋯ |
| L(s) = 1 | + 0.210·2-s − 0.577·3-s − 0.955·4-s + 1.13·5-s − 0.121·6-s − 0.377·7-s − 0.412·8-s + 0.333·9-s + 0.240·10-s + 1.85·11-s + 0.551·12-s + 1.29·13-s − 0.0796·14-s − 0.657·15-s + 0.868·16-s − 1.06·17-s + 0.0702·18-s − 1.15·19-s − 1.08·20-s + 0.218·21-s + 0.391·22-s + 1.90·23-s + 0.237·24-s + 0.297·25-s + 0.273·26-s − 0.192·27-s + 0.361·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.768878954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.768878954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 0.298T + 2T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 11 | \( 1 - 6.15T + 11T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 5.03T + 19T^{2} \) |
| 23 | \( 1 - 9.11T + 23T^{2} \) |
| 29 | \( 1 + 4.28T + 29T^{2} \) |
| 31 | \( 1 - 0.0487T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 - 1.89T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 6.07T + 53T^{2} \) |
| 59 | \( 1 + 9.37T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 5.59T + 67T^{2} \) |
| 71 | \( 1 - 7.54T + 71T^{2} \) |
| 73 | \( 1 - 5.40T + 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 0.133T + 97T^{2} \) |
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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
−9.003242161975005121508667100542, −8.484945782617781208645420838218, −6.90291419869118292121678165008, −6.38322145669723333219497003387, −5.89718059652994695017894983972, −4.94450947735057328728811409652, −4.13709668071529696553940674130, −3.43104530539583768033144342193, −1.88689773443113863851843202918, −0.888210803136682772547570392725,
0.888210803136682772547570392725, 1.88689773443113863851843202918, 3.43104530539583768033144342193, 4.13709668071529696553940674130, 4.94450947735057328728811409652, 5.89718059652994695017894983972, 6.38322145669723333219497003387, 6.90291419869118292121678165008, 8.484945782617781208645420838218, 9.003242161975005121508667100542
