| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 2·11-s + 6·13-s + 15-s − 4·17-s − 4·19-s + 21-s − 8·23-s − 4·25-s + 27-s + 2·29-s + 5·31-s + 2·33-s + 35-s − 3·37-s + 6·39-s − 6·41-s − 8·43-s + 45-s − 3·47-s − 6·49-s − 4·51-s + 7·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.66·13-s + 0.258·15-s − 0.970·17-s − 0.917·19-s + 0.218·21-s − 1.66·23-s − 4/5·25-s + 0.192·27-s + 0.371·29-s + 0.898·31-s + 0.348·33-s + 0.169·35-s − 0.493·37-s + 0.960·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s − 0.437·47-s − 6/7·49-s − 0.560·51-s + 0.961·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−15.27674768071546, −14.80900727521215, −14.13061830744288, −13.67028798150986, −13.45177613278753, −12.87453990579153, −12.02422157833544, −11.64932275809658, −11.08435068837604, −10.37956420399979, −10.02062916077611, −9.361079758092519, −8.640240909120010, −8.363699747011581, −8.010530316585747, −6.909799025177097, −6.498718639434491, −6.065920800705894, −5.322504994214969, −4.418412081352115, −3.996216668553897, −3.447553215155438, −2.467953534590007, −1.812131194500516, −1.344049507102246, 0,
1.344049507102246, 1.812131194500516, 2.467953534590007, 3.447553215155438, 3.996216668553897, 4.418412081352115, 5.322504994214969, 6.065920800705894, 6.498718639434491, 6.909799025177097, 8.010530316585747, 8.363699747011581, 8.640240909120010, 9.361079758092519, 10.02062916077611, 10.37956420399979, 11.08435068837604, 11.64932275809658, 12.02422157833544, 12.87453990579153, 13.45177613278753, 13.67028798150986, 14.13061830744288, 14.80900727521215, 15.27674768071546
