L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3729296206\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3729296206\) |
\(L(1)\) |
\(\approx\) |
\(0.3838066288\) |
\(L(1)\) |
\(\approx\) |
\(0.3838066288\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 67 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−31.80869018654966570048879528906, −30.2983684160728915396578120066, −29.03735518638643006914975182318, −28.62224652761012483734243629966, −27.2611308641117556922049508591, −26.70174569224816843100029768924, −25.25823792241517996178420996375, −23.91332809292266679012770993527, −23.089595227482036622305087852069, −21.748513425850136642643612749868, −20.24915647424801226494257358160, −19.089227625032628781940265850209, −18.3437356923891410539292266510, −16.86174789733574687551804808301, −16.14199851379843707740194080055, −15.220716962584560554285096203369, −12.68248175206236824036542572352, −11.85025250054448170584466652275, −10.59843546331941620439755468854, −9.613018248078095366246116971147, −7.760383041884180603322129774131, −6.8856313660388414445999220040, −5.27458723627780946882402310139, −3.12142075096677987543258628564, −0.60431475891438510756499008752,
0.60431475891438510756499008752, 3.12142075096677987543258628564, 5.27458723627780946882402310139, 6.8856313660388414445999220040, 7.760383041884180603322129774131, 9.613018248078095366246116971147, 10.59843546331941620439755468854, 11.85025250054448170584466652275, 12.68248175206236824036542572352, 15.220716962584560554285096203369, 16.14199851379843707740194080055, 16.86174789733574687551804808301, 18.3437356923891410539292266510, 19.089227625032628781940265850209, 20.24915647424801226494257358160, 21.748513425850136642643612749868, 23.089595227482036622305087852069, 23.91332809292266679012770993527, 25.25823792241517996178420996375, 26.70174569224816843100029768924, 27.2611308641117556922049508591, 28.62224652761012483734243629966, 29.03735518638643006914975182318, 30.2983684160728915396578120066, 31.80869018654966570048879528906