| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3·7-s + 8-s + 9-s + 11-s − 12-s + 3·14-s + 16-s − 17-s + 18-s + 6·19-s − 3·21-s + 22-s − 24-s − 5·25-s − 27-s + 3·28-s + 6·29-s + 32-s − 33-s − 34-s + 36-s − 3·37-s + 6·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.801·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.37·19-s − 0.654·21-s + 0.213·22-s − 0.204·24-s − 25-s − 0.192·27-s + 0.566·28-s + 1.11·29-s + 0.176·32-s − 0.174·33-s − 0.171·34-s + 1/6·36-s − 0.493·37-s + 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.236674450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.236674450\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.20132334992542, −12.51447336590017, −12.07158536902795, −11.74437270776835, −11.32573115833062, −11.06360312925321, −10.28542881890452, −10.00183360326675, −9.442322566695883, −8.747721351126921, −8.190262638104194, −7.814538519891722, −7.291399685217069, −6.615173254811493, −6.437771075193485, −5.508774777448050, −5.346465118161782, −4.777988213599260, −4.392666607483863, −3.672125722737478, −3.238721028773247, −2.456891028618989, −1.714456958871368, −1.385078499976491, −0.5380498039226102,
0.5380498039226102, 1.385078499976491, 1.714456958871368, 2.456891028618989, 3.238721028773247, 3.672125722737478, 4.392666607483863, 4.777988213599260, 5.346465118161782, 5.508774777448050, 6.437771075193485, 6.615173254811493, 7.291399685217069, 7.814538519891722, 8.190262638104194, 8.747721351126921, 9.442322566695883, 10.00183360326675, 10.28542881890452, 11.06360312925321, 11.32573115833062, 11.74437270776835, 12.07158536902795, 12.51447336590017, 13.20132334992542
