L(s) = 1 | + 3-s − 5-s + 9-s + 3·11-s + 13-s − 15-s + 3·17-s − 23-s + 25-s + 27-s + 6·29-s − 10·31-s + 3·33-s − 9·37-s + 39-s + 3·41-s − 10·43-s − 45-s + 2·47-s + 3·51-s + 10·53-s − 3·55-s − 9·59-s − 2·61-s − 65-s − 5·67-s − 69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.258·15-s + 0.727·17-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.79·31-s + 0.522·33-s − 1.47·37-s + 0.160·39-s + 0.468·41-s − 1.52·43-s − 0.149·45-s + 0.291·47-s + 0.420·51-s + 1.37·53-s − 0.404·55-s − 1.17·59-s − 0.256·61-s − 0.124·65-s − 0.610·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.805686221\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.805686221\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.06273083255083, −13.64509874581868, −13.12387672553096, −12.40540071858056, −12.13000456798736, −11.68053782490753, −11.02809389701182, −10.46431349897113, −10.07796894749518, −9.372397896276382, −8.919292074491267, −8.541725603973013, −7.965433499212240, −7.344645347311549, −6.992976661804367, −6.351436153389929, −5.735702289304416, −5.092691247178868, −4.456037121956513, −3.806649877429641, −3.433424569259517, −2.866148702503693, −1.890014623133373, −1.446623306364018, −0.5364096017922856,
0.5364096017922856, 1.446623306364018, 1.890014623133373, 2.866148702503693, 3.433424569259517, 3.806649877429641, 4.456037121956513, 5.092691247178868, 5.735702289304416, 6.351436153389929, 6.992976661804367, 7.344645347311549, 7.965433499212240, 8.541725603973013, 8.919292074491267, 9.372397896276382, 10.07796894749518, 10.46431349897113, 11.02809389701182, 11.68053782490753, 12.13000456798736, 12.40540071858056, 13.12387672553096, 13.64509874581868, 14.06273083255083