L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 11-s + 5·13-s + 14-s + 16-s − 4·17-s + 7·19-s + 22-s + 5·26-s + 28-s − 6·29-s + 32-s − 4·34-s − 4·37-s + 7·38-s − 3·41-s + 11·43-s + 44-s + 47-s + 49-s + 5·52-s + 53-s + 56-s − 6·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.301·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 1.60·19-s + 0.213·22-s + 0.980·26-s + 0.188·28-s − 1.11·29-s + 0.176·32-s − 0.685·34-s − 0.657·37-s + 1.13·38-s − 0.468·41-s + 1.67·43-s + 0.150·44-s + 0.145·47-s + 1/7·49-s + 0.693·52-s + 0.137·53-s + 0.133·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.132585870\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.132585870\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.69281721002622315616091221974, −6.80845046589559813447482285302, −6.37864901793928866415289483412, −5.42686166362065127311771623046, −5.14390592081714508072053021669, −3.91607667688486460192432538112, −3.77786565625704649931489889900, −2.70112667558685491883558397688, −1.79599909564359746541011864137, −0.920612456787474847026333699856,
0.920612456787474847026333699856, 1.79599909564359746541011864137, 2.70112667558685491883558397688, 3.77786565625704649931489889900, 3.91607667688486460192432538112, 5.14390592081714508072053021669, 5.42686166362065127311771623046, 6.37864901793928866415289483412, 6.80845046589559813447482285302, 7.69281721002622315616091221974