| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s − 7-s + 8-s + 9-s − 3·10-s + 5·11-s + 12-s − 14-s − 3·15-s + 16-s − 3·17-s + 18-s − 19-s − 3·20-s − 21-s + 5·22-s + 23-s + 24-s + 4·25-s + 27-s − 28-s + 5·29-s − 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.50·11-s + 0.288·12-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.229·19-s − 0.670·20-s − 0.218·21-s + 1.06·22-s + 0.208·23-s + 0.204·24-s + 4/5·25-s + 0.192·27-s − 0.188·28-s + 0.928·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.177368100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.177368100\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−7.79451205432877932606201632081, −7.22294398015253099317782186378, −6.55487529290544743886442486730, −6.01780029173144062292882558186, −4.61616996002906401558521058978, −4.37211697575151349823083452182, −3.58311886156486752959928440676, −3.07023171611208217149652986410, −1.99193245524869566548268073668, −0.795853244729407449928566082408,
0.795853244729407449928566082408, 1.99193245524869566548268073668, 3.07023171611208217149652986410, 3.58311886156486752959928440676, 4.37211697575151349823083452182, 4.61616996002906401558521058978, 6.01780029173144062292882558186, 6.55487529290544743886442486730, 7.22294398015253099317782186378, 7.79451205432877932606201632081
