| L(s) = 1 | − 1.87·2-s + 1.53·4-s − 0.879·5-s − 2.87·7-s + 0.879·8-s + 1.65·10-s + 1.83·11-s − 2.75·13-s + 5.41·14-s − 4.71·16-s + 7.10·17-s − 1.34·20-s − 3.45·22-s − 6.59·23-s − 4.22·25-s + 5.18·26-s − 4.41·28-s + 3.12·29-s + 7.65·31-s + 7.10·32-s − 13.3·34-s + 2.53·35-s + 2.83·37-s − 0.773·40-s − 3.98·41-s − 11.4·43-s + 2.81·44-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.766·4-s − 0.393·5-s − 1.08·7-s + 0.310·8-s + 0.522·10-s + 0.554·11-s − 0.765·13-s + 1.44·14-s − 1.17·16-s + 1.72·17-s − 0.301·20-s − 0.736·22-s − 1.37·23-s − 0.845·25-s + 1.01·26-s − 0.833·28-s + 0.580·29-s + 1.37·31-s + 1.25·32-s − 2.29·34-s + 0.428·35-s + 0.466·37-s − 0.122·40-s − 0.622·41-s − 1.74·43-s + 0.424·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5055071220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5055071220\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 5 | \( 1 + 0.879T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 + 2.75T + 13T^{2} \) |
| 17 | \( 1 - 7.10T + 17T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 - 7.65T + 31T^{2} \) |
| 37 | \( 1 - 2.83T + 37T^{2} \) |
| 41 | \( 1 + 3.98T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 - 8.41T + 59T^{2} \) |
| 61 | \( 1 - 0.615T + 61T^{2} \) |
| 67 | \( 1 + 3.67T + 67T^{2} \) |
| 71 | \( 1 - 7.45T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 1.61T + 79T^{2} \) |
| 83 | \( 1 + 0.985T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 + 5.90T + 97T^{2} \) |
| 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
−8.525203823057253548884686895379, −8.086292985959810148956976333675, −7.37543056755714803278995892556, −6.64403946817044281969703312610, −5.90722834186597376930563851335, −4.75846680606184486605457457531, −3.80270961804269768889853734081, −2.93978745038615739041312828548, −1.70998649748166123457238547730, −0.52372169175375729920119235283,
0.52372169175375729920119235283, 1.70998649748166123457238547730, 2.93978745038615739041312828548, 3.80270961804269768889853734081, 4.75846680606184486605457457531, 5.90722834186597376930563851335, 6.64403946817044281969703312610, 7.37543056755714803278995892556, 8.086292985959810148956976333675, 8.525203823057253548884686895379
