L(s) = 1 | − 3-s − 5-s + 5·7-s − 2·9-s − 3·11-s − 2·13-s + 15-s − 4·17-s + 4·19-s − 5·21-s + 2·23-s + 25-s + 5·27-s + 2·29-s + 3·33-s − 5·35-s − 37-s + 2·39-s + 7·41-s + 10·43-s + 2·45-s − 11·47-s + 18·49-s + 4·51-s − 3·53-s + 3·55-s − 4·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.88·7-s − 2/3·9-s − 0.904·11-s − 0.554·13-s + 0.258·15-s − 0.970·17-s + 0.917·19-s − 1.09·21-s + 0.417·23-s + 1/5·25-s + 0.962·27-s + 0.371·29-s + 0.522·33-s − 0.845·35-s − 0.164·37-s + 0.320·39-s + 1.09·41-s + 1.52·43-s + 0.298·45-s − 1.60·47-s + 18/7·49-s + 0.560·51-s − 0.412·53-s + 0.404·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.350168849\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.350168849\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.605758983200435886333909864724, −7.85898391187582192436857946962, −7.52400418007960556062901819378, −6.43126482720515848183226026435, −5.39765104085023336200732515031, −4.97488125379322241916877032519, −4.34970436039036781131134049365, −2.96706194794650080029042292641, −2.06114621236543456699155934711, −0.72841160829155423220093703072,
0.72841160829155423220093703072, 2.06114621236543456699155934711, 2.96706194794650080029042292641, 4.34970436039036781131134049365, 4.97488125379322241916877032519, 5.39765104085023336200732515031, 6.43126482720515848183226026435, 7.52400418007960556062901819378, 7.85898391187582192436857946962, 8.605758983200435886333909864724