| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s − 11-s − 12-s − 2·15-s + 16-s − 17-s − 18-s + 4·19-s + 2·20-s + 22-s − 4·23-s + 24-s − 25-s − 27-s + 2·30-s − 2·31-s − 32-s + 33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.365·30-s − 0.359·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.41474985844287, −12.75448464103567, −12.38362591082271, −11.82832886998992, −11.36478188010124, −11.04589051821943, −10.32073109610885, −9.953606319599769, −9.830002113359317, −9.094228742084883, −8.740014041131913, −8.006457244607499, −7.686613327895664, −7.079515713653658, −6.526514802539746, −6.145656308739706, −5.653125228849999, −5.090306839570642, −4.744417823825582, −3.771706147771581, −3.351290495690680, −2.559880452415753, −1.918858419503089, −1.581639364973890, −0.7312405146227224, 0,
0.7312405146227224, 1.581639364973890, 1.918858419503089, 2.559880452415753, 3.351290495690680, 3.771706147771581, 4.744417823825582, 5.090306839570642, 5.653125228849999, 6.145656308739706, 6.526514802539746, 7.079515713653658, 7.686613327895664, 8.006457244607499, 8.740014041131913, 9.094228742084883, 9.830002113359317, 9.953606319599769, 10.32073109610885, 11.04589051821943, 11.36478188010124, 11.82832886998992, 12.38362591082271, 12.75448464103567, 13.41474985844287
