| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s − 3·9-s + 10-s − 11-s − 2·13-s + 16-s + 2·17-s + 3·18-s − 19-s − 20-s + 22-s − 4·23-s + 25-s + 2·26-s + 6·29-s − 32-s − 2·34-s − 3·36-s + 6·37-s + 38-s + 40-s − 2·41-s − 4·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s − 9-s + 0.316·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.229·19-s − 0.223·20-s + 0.213·22-s − 0.834·23-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 0.176·32-s − 0.342·34-s − 1/2·36-s + 0.986·37-s + 0.162·38-s + 0.158·40-s − 0.312·41-s − 0.609·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4155486060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4155486060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.90128958852783, −13.14994635913706, −12.65774600708490, −12.03542950210245, −11.81436292650865, −11.27055618820711, −10.77799670630606, −10.21502799490800, −9.846768545855318, −9.238403442838116, −8.735708798533929, −8.090080913474094, −7.977700810688329, −7.380946289835382, −6.671199303479813, −6.191406628251423, −5.730671432344080, −4.946451416208027, −4.578317417614222, −3.687419577163712, −3.094084004766165, −2.643462385951459, −1.948660453708746, −1.128914977081646, −0.2426035963777778,
0.2426035963777778, 1.128914977081646, 1.948660453708746, 2.643462385951459, 3.094084004766165, 3.687419577163712, 4.578317417614222, 4.946451416208027, 5.730671432344080, 6.191406628251423, 6.671199303479813, 7.380946289835382, 7.977700810688329, 8.090080913474094, 8.735708798533929, 9.238403442838116, 9.846768545855318, 10.21502799490800, 10.77799670630606, 11.27055618820711, 11.81436292650865, 12.03542950210245, 12.65774600708490, 13.14994635913706, 13.90128958852783
