L(s) = 1 | − 3-s + 7-s + 9-s − 5·11-s − 4·13-s − 2·17-s + 4·19-s − 21-s − 9·23-s − 27-s − 5·29-s + 2·31-s + 5·33-s + 11·37-s + 4·39-s + 8·41-s + 11·43-s − 12·47-s + 49-s + 2·51-s + 10·53-s − 4·57-s + 12·59-s − 12·61-s + 63-s − 3·67-s + 9·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 1.10·13-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 1.87·23-s − 0.192·27-s − 0.928·29-s + 0.359·31-s + 0.870·33-s + 1.80·37-s + 0.640·39-s + 1.24·41-s + 1.67·43-s − 1.75·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s − 0.529·57-s + 1.56·59-s − 1.53·61-s + 0.125·63-s − 0.366·67-s + 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.019023889\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.019023889\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−7.959792450498421239488237713435, −7.86417655869103773376058145043, −7.05990049796948690191465038733, −5.98083330409592196877082054154, −5.50115047993490802356958886125, −4.72713583356704746215466910248, −4.04441923146332234595676426489, −2.71653407675847405555652329714, −2.08269644123298725038783595280, −0.56380085934414660125932016351,
0.56380085934414660125932016351, 2.08269644123298725038783595280, 2.71653407675847405555652329714, 4.04441923146332234595676426489, 4.72713583356704746215466910248, 5.50115047993490802356958886125, 5.98083330409592196877082054154, 7.05990049796948690191465038733, 7.86417655869103773376058145043, 7.959792450498421239488237713435