L(s) = 1 | − 5-s + 7-s + 2·11-s − 2·13-s + 6·17-s − 6·23-s + 25-s − 35-s − 6·37-s + 2·41-s + 12·43-s + 12·47-s + 49-s − 2·55-s + 4·59-s − 10·61-s + 2·65-s + 4·67-s + 10·71-s − 14·73-s + 2·77-s − 6·85-s + 2·89-s − 2·91-s + 10·97-s − 2·101-s − 12·103-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.603·11-s − 0.554·13-s + 1.45·17-s − 1.25·23-s + 1/5·25-s − 0.169·35-s − 0.986·37-s + 0.312·41-s + 1.82·43-s + 1.75·47-s + 1/7·49-s − 0.269·55-s + 0.520·59-s − 1.28·61-s + 0.248·65-s + 0.488·67-s + 1.18·71-s − 1.63·73-s + 0.227·77-s − 0.650·85-s + 0.211·89-s − 0.209·91-s + 1.01·97-s − 0.199·101-s − 1.18·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.872570172\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.872570172\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.080181096969844331142441770200, −7.59540748232831461229574068857, −6.96185217964779934884547946892, −5.93487964405357589138383346022, −5.42669252171585660477901551773, −4.39148600441151057366175690253, −3.84934431047133246191699656421, −2.90071357559009479174841412700, −1.86012418634430356429804219261, −0.76186670716461730787425627720,
0.76186670716461730787425627720, 1.86012418634430356429804219261, 2.90071357559009479174841412700, 3.84934431047133246191699656421, 4.39148600441151057366175690253, 5.42669252171585660477901551773, 5.93487964405357589138383346022, 6.96185217964779934884547946892, 7.59540748232831461229574068857, 8.080181096969844331142441770200