| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 2·11-s − 13-s + 16-s + 3·17-s + 19-s − 2·20-s − 2·22-s − 25-s + 26-s + 3·29-s + 9·31-s − 32-s − 3·34-s − 4·37-s − 38-s + 2·40-s − 7·41-s + 5·43-s + 2·44-s + 8·47-s + 50-s − 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 0.603·11-s − 0.277·13-s + 1/4·16-s + 0.727·17-s + 0.229·19-s − 0.447·20-s − 0.426·22-s − 1/5·25-s + 0.196·26-s + 0.557·29-s + 1.61·31-s − 0.176·32-s − 0.514·34-s − 0.657·37-s − 0.162·38-s + 0.316·40-s − 1.09·41-s + 0.762·43-s + 0.301·44-s + 1.16·47-s + 0.141·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.182670817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182670817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.57027425278183, −12.55306619605816, −12.25282657056484, −11.91146897906695, −11.51844073415334, −10.98682948980828, −10.36139154964691, −10.02847580359400, −9.549929143124009, −8.928115140921150, −8.464294985485283, −8.082124576763183, −7.565764459556073, −7.052119098452377, −6.705805413779287, −5.966717356838089, −5.523053079958867, −4.788554860575675, −4.163057156985168, −3.786730500496721, −2.969390933011863, −2.660254338550877, −1.662462707593865, −1.117755304327842, −0.4066287557131318,
0.4066287557131318, 1.117755304327842, 1.662462707593865, 2.660254338550877, 2.969390933011863, 3.786730500496721, 4.163057156985168, 4.788554860575675, 5.523053079958867, 5.966717356838089, 6.705805413779287, 7.052119098452377, 7.565764459556073, 8.082124576763183, 8.464294985485283, 8.928115140921150, 9.549929143124009, 10.02847580359400, 10.36139154964691, 10.98682948980828, 11.51844073415334, 11.91146897906695, 12.25282657056484, 12.55306619605816, 13.57027425278183
