| L(s) = 1 | + 2-s − 2·3-s − 7·4-s − 2·6-s − 15·8-s − 23·9-s − 8·11-s + 14·12-s + 28·13-s + 41·16-s + 54·17-s − 23·18-s + 110·19-s − 8·22-s − 48·23-s + 30·24-s + 28·26-s + 100·27-s − 110·29-s − 12·31-s + 161·32-s + 16·33-s + 54·34-s + 161·36-s + 246·37-s + 110·38-s − 56·39-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.384·3-s − 7/8·4-s − 0.136·6-s − 0.662·8-s − 0.851·9-s − 0.219·11-s + 0.336·12-s + 0.597·13-s + 0.640·16-s + 0.770·17-s − 0.301·18-s + 1.32·19-s − 0.0775·22-s − 0.435·23-s + 0.255·24-s + 0.211·26-s + 0.712·27-s − 0.704·29-s − 0.0695·31-s + 0.889·32-s + 0.0844·33-s + 0.272·34-s + 0.745·36-s + 1.09·37-s + 0.469·38-s − 0.229·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 110 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 110 T + p^{3} T^{2} \) |
| 31 | \( 1 + 12 T + p^{3} T^{2} \) |
| 37 | \( 1 - 246 T + p^{3} T^{2} \) |
| 41 | \( 1 + 182 T + p^{3} T^{2} \) |
| 43 | \( 1 + 128 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 810 T + p^{3} T^{2} \) |
| 61 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 244 T + p^{3} T^{2} \) |
| 71 | \( 1 + 768 T + p^{3} T^{2} \) |
| 73 | \( 1 + 702 T + p^{3} T^{2} \) |
| 79 | \( 1 - 440 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1302 T + p^{3} T^{2} \) |
| 89 | \( 1 + 730 T + p^{3} T^{2} \) |
| 97 | \( 1 - 294 T + p^{3} T^{2} \) |
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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
−8.953384588572920504375840476603, −8.197180197794439656730769342609, −7.37173749140732420786463646080, −5.98195015228847581908013282634, −5.63447974264080229985561161946, −4.74709078617697647876790167589, −3.67309813869389593628344148345, −2.89266512499027057938802213026, −1.15098830882669493082327319620, 0,
1.15098830882669493082327319620, 2.89266512499027057938802213026, 3.67309813869389593628344148345, 4.74709078617697647876790167589, 5.63447974264080229985561161946, 5.98195015228847581908013282634, 7.37173749140732420786463646080, 8.197180197794439656730769342609, 8.953384588572920504375840476603
