| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 5·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s + 10·10-s + 12·11-s − 12·12-s + 2·13-s − 14·14-s + 15·15-s + 16·16-s − 18·17-s − 18·18-s + 56·19-s − 20·20-s − 21·21-s − 24·22-s − 156·23-s + 24·24-s + 25·25-s − 4·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.328·11-s − 0.288·12-s + 0.0426·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.256·17-s − 0.235·18-s + 0.676·19-s − 0.223·20-s − 0.218·21-s − 0.232·22-s − 1.41·23-s + 0.204·24-s + 1/5·25-s − 0.0301·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + 156 T + p^{3} T^{2} \) |
| 29 | \( 1 + 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 52 T + p^{3} T^{2} \) |
| 37 | \( 1 + 178 T + p^{3} T^{2} \) |
| 41 | \( 1 + 138 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 456 T + p^{3} T^{2} \) |
| 53 | \( 1 + 198 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 - 110 T + p^{3} T^{2} \) |
| 67 | \( 1 + 196 T + p^{3} T^{2} \) |
| 71 | \( 1 + 936 T + p^{3} T^{2} \) |
| 73 | \( 1 - 542 T + p^{3} T^{2} \) |
| 79 | \( 1 - 992 T + p^{3} T^{2} \) |
| 83 | \( 1 + 276 T + p^{3} T^{2} \) |
| 89 | \( 1 - 630 T + p^{3} T^{2} \) |
| 97 | \( 1 - 110 T + p^{3} T^{2} \) |
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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
−11.50349896476997479582438801378, −10.47295122931231392990389360056, −9.531357922044252908352881557464, −8.376305907423197518454620966567, −7.45702679743899948770516904639, −6.39580326601707841263006404860, −5.13927132311198187264103254368, −3.67073875116458922916875141602, −1.69965255347246301006725902367, 0,
1.69965255347246301006725902367, 3.67073875116458922916875141602, 5.13927132311198187264103254368, 6.39580326601707841263006404860, 7.45702679743899948770516904639, 8.376305907423197518454620966567, 9.531357922044252908352881557464, 10.47295122931231392990389360056, 11.50349896476997479582438801378
