| L(s) = 1 | − 1.77·2-s − 3-s + 1.13·4-s + 1.38·5-s + 1.77·6-s + 7-s + 1.53·8-s + 9-s − 2.45·10-s + 3.51·11-s − 1.13·12-s + 4.71·13-s − 1.77·14-s − 1.38·15-s − 4.98·16-s − 4.68·17-s − 1.77·18-s − 3.74·19-s + 1.57·20-s − 21-s − 6.22·22-s − 2.00·23-s − 1.53·24-s − 3.08·25-s − 8.34·26-s − 27-s + 1.13·28-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.577·3-s + 0.567·4-s + 0.618·5-s + 0.722·6-s + 0.377·7-s + 0.541·8-s + 0.333·9-s − 0.774·10-s + 1.05·11-s − 0.327·12-s + 1.30·13-s − 0.473·14-s − 0.357·15-s − 1.24·16-s − 1.13·17-s − 0.417·18-s − 0.859·19-s + 0.351·20-s − 0.218·21-s − 1.32·22-s − 0.417·23-s − 0.312·24-s − 0.617·25-s − 1.63·26-s − 0.192·27-s + 0.214·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 5 | \( 1 - 1.38T + 5T^{2} \) |
| 11 | \( 1 - 3.51T + 11T^{2} \) |
| 13 | \( 1 - 4.71T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 + 2.00T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 0.875T + 31T^{2} \) |
| 37 | \( 1 + 9.02T + 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 + 2.16T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 0.801T + 59T^{2} \) |
| 61 | \( 1 - 3.10T + 61T^{2} \) |
| 67 | \( 1 - 6.63T + 67T^{2} \) |
| 71 | \( 1 + 8.91T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 3.05T + 79T^{2} \) |
| 83 | \( 1 - 4.04T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 8.60T + 97T^{2} \) |
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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
−8.533445306149136611547444091952, −8.031397669763080101083131168208, −6.78197036340811529342582327555, −6.50578136182583830807558518535, −5.54192523581471887056096295296, −4.49318108275252003760356303380, −3.74489094271938583516080656130, −1.92566336695622079396720915850, −1.47584926787718571749700258161, 0,
1.47584926787718571749700258161, 1.92566336695622079396720915850, 3.74489094271938583516080656130, 4.49318108275252003760356303380, 5.54192523581471887056096295296, 6.50578136182583830807558518535, 6.78197036340811529342582327555, 8.031397669763080101083131168208, 8.533445306149136611547444091952
