| L(s) = 1 | + 3·3-s + 7-s + 6·9-s − 3·11-s − 13-s + 17-s + 4·19-s + 3·21-s − 4·23-s + 9·27-s − 9·29-s + 6·31-s − 9·33-s + 8·37-s − 3·39-s + 6·41-s + 8·43-s + 7·47-s + 49-s + 3·51-s + 8·53-s + 12·57-s + 4·59-s + 10·61-s + 6·63-s + 8·67-s − 12·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.377·7-s + 2·9-s − 0.904·11-s − 0.277·13-s + 0.242·17-s + 0.917·19-s + 0.654·21-s − 0.834·23-s + 1.73·27-s − 1.67·29-s + 1.07·31-s − 1.56·33-s + 1.31·37-s − 0.480·39-s + 0.937·41-s + 1.21·43-s + 1.02·47-s + 1/7·49-s + 0.420·51-s + 1.09·53-s + 1.58·57-s + 0.520·59-s + 1.28·61-s + 0.755·63-s + 0.977·67-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.083364725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.083364725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.987184132503805995717897364380, −7.69806051465811372694172032573, −7.15018724888124145418636776853, −5.93235552604519188689122663708, −5.18213069919261638310400163682, −4.18050679297164594507333025217, −3.66056608704176785522632511267, −2.54590429894924234271990154143, −2.33245281257879554013579568769, −1.02544900745500368460829410531,
1.02544900745500368460829410531, 2.33245281257879554013579568769, 2.54590429894924234271990154143, 3.66056608704176785522632511267, 4.18050679297164594507333025217, 5.18213069919261638310400163682, 5.93235552604519188689122663708, 7.15018724888124145418636776853, 7.69806051465811372694172032573, 7.987184132503805995717897364380
