| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 0.622·7-s − 8-s + 9-s + 4.42·11-s + 12-s − 1.37·13-s + 0.622·14-s + 16-s − 6.42·17-s − 18-s − 1.37·19-s − 0.622·21-s − 4.42·22-s − 23-s − 24-s + 1.37·26-s + 27-s − 0.622·28-s − 4.23·29-s − 4·31-s − 32-s + 4.42·33-s + 6.42·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.235·7-s − 0.353·8-s + 0.333·9-s + 1.33·11-s + 0.288·12-s − 0.382·13-s + 0.166·14-s + 0.250·16-s − 1.55·17-s − 0.235·18-s − 0.316·19-s − 0.135·21-s − 0.944·22-s − 0.208·23-s − 0.204·24-s + 0.270·26-s + 0.192·27-s − 0.117·28-s − 0.786·29-s − 0.718·31-s − 0.176·32-s + 0.770·33-s + 1.10·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 0.622T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 + 6.42T + 17T^{2} \) |
| 19 | \( 1 + 1.37T + 19T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 3.37T + 53T^{2} \) |
| 59 | \( 1 - 9.61T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 6.99T + 71T^{2} \) |
| 73 | \( 1 + 2.75T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 - 2.75T + 89T^{2} \) |
| 97 | \( 1 - 2.13T + 97T^{2} \) |
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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
−8.559359405928602889981414889651, −7.43676034073385431128546156090, −6.87638009373953689059645306842, −6.32297653828020260530502818724, −5.18873584588930821924570432981, −4.11315054453020328279282118304, −3.47077496089904345466788332906, −2.28196805559722431665587707675, −1.58437240387044442982921760080, 0,
1.58437240387044442982921760080, 2.28196805559722431665587707675, 3.47077496089904345466788332906, 4.11315054453020328279282118304, 5.18873584588930821924570432981, 6.32297653828020260530502818724, 6.87638009373953689059645306842, 7.43676034073385431128546156090, 8.559359405928602889981414889651
