| L(s) = 1 | − 2-s − 2·3-s + 4-s − 5-s + 2·6-s − 7-s − 8-s + 9-s + 10-s + 6·11-s − 2·12-s − 13-s + 14-s + 2·15-s + 16-s − 18-s − 7·19-s − 20-s + 2·21-s − 6·22-s + 2·24-s + 25-s + 26-s + 4·27-s − 28-s − 29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.577·12-s − 0.277·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.235·18-s − 1.60·19-s − 0.223·20-s + 0.436·21-s − 1.27·22-s + 0.408·24-s + 1/5·25-s + 0.196·26-s + 0.769·27-s − 0.188·28-s − 0.185·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4526924316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4526924316\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.87593926634181, −14.31014507480758, −13.94820249587010, −12.76355615735655, −12.52599024110846, −12.19004596251393, −11.40445149075457, −11.20086472315106, −10.69345525813669, −10.15386479099326, −9.369663780159265, −8.972947093940266, −8.622264254731103, −7.598787359060784, −7.297657893982511, −6.510903552430660, −6.143854036023519, −5.864920589178586, −4.745269160642162, −4.320841136699365, −3.658026012004181, −2.847423985305483, −1.908470332895752, −1.170442836353396, −0.3306156484937494,
0.3306156484937494, 1.170442836353396, 1.908470332895752, 2.847423985305483, 3.658026012004181, 4.320841136699365, 4.745269160642162, 5.864920589178586, 6.143854036023519, 6.510903552430660, 7.297657893982511, 7.598787359060784, 8.622264254731103, 8.972947093940266, 9.369663780159265, 10.15386479099326, 10.69345525813669, 11.20086472315106, 11.40445149075457, 12.19004596251393, 12.52599024110846, 12.76355615735655, 13.94820249587010, 14.31014507480758, 14.87593926634181
