| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s + 9-s − 10-s + 11-s − 12-s − 4·13-s + 2·14-s + 15-s + 16-s + 18-s − 6·19-s − 20-s − 2·21-s + 22-s + 2·23-s − 24-s + 25-s − 4·26-s − 27-s + 2·28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 1.10·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.436·21-s + 0.213·22-s + 0.417·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.377·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.16357281455903, −13.56861651358802, −12.83285066116451, −12.60285455950181, −12.17045851502454, −11.53630472902086, −11.27241342742649, −10.85077602334911, −10.04847065560890, −9.948498361943053, −8.954959649973485, −8.506174505048323, −7.882167097111369, −7.473322536918082, −6.839311743629165, −6.389007726502645, −5.914831049909638, −5.084167010341817, −4.759676960217147, −4.360286545086317, −3.819856903163930, −2.888517243736784, −2.448383457448297, −1.662067406024687, −0.9319188915359832, 0,
0.9319188915359832, 1.662067406024687, 2.448383457448297, 2.888517243736784, 3.819856903163930, 4.360286545086317, 4.759676960217147, 5.084167010341817, 5.914831049909638, 6.389007726502645, 6.839311743629165, 7.473322536918082, 7.882167097111369, 8.506174505048323, 8.954959649973485, 9.948498361943053, 10.04847065560890, 10.85077602334911, 11.27241342742649, 11.53630472902086, 12.17045851502454, 12.60285455950181, 12.83285066116451, 13.56861651358802, 14.16357281455903
