| L(s) = 1 | + 30.2·3-s − 141.·7-s + 670.·9-s − 273.·11-s − 945.·13-s − 1.21e3·17-s − 135.·19-s − 4.27e3·21-s − 2.73e3·23-s + 1.29e4·27-s − 3.07e3·29-s + 1.41e3·31-s − 8.26e3·33-s + 3.15e3·37-s − 2.85e4·39-s − 1.34e4·41-s + 5.43e3·43-s − 3.26e3·47-s + 3.17e3·49-s − 3.66e4·51-s − 2.14e4·53-s − 4.10e3·57-s + 4.14e4·59-s − 2.43e4·61-s − 9.47e4·63-s − 3.23e4·67-s − 8.25e4·69-s + ⋯ |
| L(s) = 1 | + 1.93·3-s − 1.09·7-s + 2.75·9-s − 0.681·11-s − 1.55·13-s − 1.01·17-s − 0.0863·19-s − 2.11·21-s − 1.07·23-s + 3.41·27-s − 0.679·29-s + 0.263·31-s − 1.32·33-s + 0.378·37-s − 3.00·39-s − 1.25·41-s + 0.448·43-s − 0.215·47-s + 0.188·49-s − 1.97·51-s − 1.04·53-s − 0.167·57-s + 1.54·59-s − 0.838·61-s − 3.00·63-s − 0.879·67-s − 2.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 30.2T + 243T^{2} \) |
| 7 | \( 1 + 141.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 273.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 945.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.21e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 135.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.73e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.15e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.43e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.26e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.14e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.43e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.86e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.08e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.68e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.743119765948505741168653436246, −9.194410522219928394969664716935, −8.154823723438107718582891575551, −7.43092177667232257446274584876, −6.54369233067865337821480519154, −4.79029901663408179373286700273, −3.71200234586889404540471896710, −2.72927127657285763448966296137, −2.04877654352521529211727535922, 0,
2.04877654352521529211727535922, 2.72927127657285763448966296137, 3.71200234586889404540471896710, 4.79029901663408179373286700273, 6.54369233067865337821480519154, 7.43092177667232257446274584876, 8.154823723438107718582891575551, 9.194410522219928394969664716935, 9.743119765948505741168653436246
