| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s − 7-s + 9-s + 2·10-s − 4·11-s + 2·12-s − 13-s − 2·14-s + 15-s − 4·16-s + 5·17-s + 2·18-s + 6·19-s + 2·20-s − 21-s − 8·22-s + 25-s − 2·26-s + 27-s − 2·28-s + 9·29-s + 2·30-s + 4·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.577·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s − 16-s + 1.21·17-s + 0.471·18-s + 1.37·19-s + 0.447·20-s − 0.218·21-s − 1.70·22-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.377·28-s + 1.67·29-s + 0.365·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.737702737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.737702737\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−14.27677371564345, −13.89608657175962, −13.37749694886587, −13.13441051750936, −12.44763320849486, −12.06404496343841, −11.69920923947347, −10.81869165281865, −10.20521063762645, −9.876889619765429, −9.388301257577220, −8.537261148503494, −8.157564585507837, −7.405630950802590, −6.965554352882989, −6.320992668354111, −5.579111968695776, −5.350318947832184, −4.739827816015866, −4.142630971090056, −3.204050971091217, −3.048471956512508, −2.550338575865937, −1.631057880146558, −0.6675349336836407,
0.6675349336836407, 1.631057880146558, 2.550338575865937, 3.048471956512508, 3.204050971091217, 4.142630971090056, 4.739827816015866, 5.350318947832184, 5.579111968695776, 6.320992668354111, 6.965554352882989, 7.405630950802590, 8.157564585507837, 8.537261148503494, 9.388301257577220, 9.876889619765429, 10.20521063762645, 10.81869165281865, 11.69920923947347, 12.06404496343841, 12.44763320849486, 13.13441051750936, 13.37749694886587, 13.89608657175962, 14.27677371564345
