L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 3·7-s + 9-s + 2·10-s − 11-s − 2·12-s − 13-s + 6·14-s − 15-s − 4·16-s − 17-s + 2·18-s − 2·19-s + 2·20-s − 3·21-s − 2·22-s − 3·23-s + 25-s − 2·26-s − 27-s + 6·28-s − 2·29-s − 2·30-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1.13·7-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.577·12-s − 0.277·13-s + 1.60·14-s − 0.258·15-s − 16-s − 0.242·17-s + 0.471·18-s − 0.458·19-s + 0.447·20-s − 0.654·21-s − 0.426·22-s − 0.625·23-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 1.13·28-s − 0.371·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.121203277\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.121203277\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−12.71270080469208063736551190005, −11.61771491043825325349583447333, −11.06044491519637715763621834009, −9.771886360481556531305754596541, −8.335023576107244243788704360049, −6.97057067834936316995403518099, −5.79375120182502147608870430489, −5.05428953135960136559927317103, −4.05163184530249924926116818344, −2.21108906834861592943657431933,
2.21108906834861592943657431933, 4.05163184530249924926116818344, 5.05428953135960136559927317103, 5.79375120182502147608870430489, 6.97057067834936316995403518099, 8.335023576107244243788704360049, 9.771886360481556531305754596541, 11.06044491519637715763621834009, 11.61771491043825325349583447333, 12.71270080469208063736551190005