L(s) = 1 | − 3-s − 7-s + 9-s + 11-s + 3·13-s − 17-s − 5·19-s + 21-s − 4·23-s − 27-s + 10·31-s − 33-s + 5·37-s − 3·39-s − 3·41-s − 10·43-s + 2·47-s + 49-s + 51-s + 13·53-s + 5·57-s − 6·59-s − 61-s − 63-s − 67-s + 4·69-s + 7·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.832·13-s − 0.242·17-s − 1.14·19-s + 0.218·21-s − 0.834·23-s − 0.192·27-s + 1.79·31-s − 0.174·33-s + 0.821·37-s − 0.480·39-s − 0.468·41-s − 1.52·43-s + 0.291·47-s + 1/7·49-s + 0.140·51-s + 1.78·53-s + 0.662·57-s − 0.781·59-s − 0.128·61-s − 0.125·63-s − 0.122·67-s + 0.481·69-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.741746374\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.741746374\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.32819543881247, −12.07712420130287, −11.61024583358424, −11.22117508818726, −10.62886796528719, −10.26944957142833, −9.963287284430616, −9.345924593742208, −8.817516281666365, −8.411201804180931, −8.010261211844804, −7.420443377087953, −6.730484164214887, −6.426417017902574, −6.147962391536436, −5.630680432042815, −4.918399163723590, −4.522894865076789, −3.963069197010515, −3.593581326176381, −2.867484361331718, −2.256608502762887, −1.704425534860027, −0.9737394619070005, −0.4099768395616594,
0.4099768395616594, 0.9737394619070005, 1.704425534860027, 2.256608502762887, 2.867484361331718, 3.593581326176381, 3.963069197010515, 4.522894865076789, 4.918399163723590, 5.630680432042815, 6.147962391536436, 6.426417017902574, 6.730484164214887, 7.420443377087953, 8.010261211844804, 8.411201804180931, 8.817516281666365, 9.345924593742208, 9.963287284430616, 10.26944957142833, 10.62886796528719, 11.22117508818726, 11.61024583358424, 12.07712420130287, 12.32819543881247