L(s) = 1 | + 7-s + 4·11-s − 3·13-s + 4·17-s − 19-s + 8·29-s + 31-s + 2·37-s − 2·41-s − 11·43-s + 2·47-s − 6·49-s + 10·53-s + 6·59-s + 11·61-s + 9·67-s − 6·71-s − 14·73-s + 4·77-s + 16·79-s − 2·83-s − 3·91-s − 11·97-s − 14·101-s + 8·103-s − 6·107-s − 11·109-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 1.20·11-s − 0.832·13-s + 0.970·17-s − 0.229·19-s + 1.48·29-s + 0.179·31-s + 0.328·37-s − 0.312·41-s − 1.67·43-s + 0.291·47-s − 6/7·49-s + 1.37·53-s + 0.781·59-s + 1.40·61-s + 1.09·67-s − 0.712·71-s − 1.63·73-s + 0.455·77-s + 1.80·79-s − 0.219·83-s − 0.314·91-s − 1.11·97-s − 1.39·101-s + 0.788·103-s − 0.580·107-s − 1.05·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.343548519\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.343548519\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.119344141437620494656346966862, −7.02018422312595018715997814354, −6.74170906319219068322168674229, −5.78867423767551983272336508703, −5.06645566925151233570382408451, −4.37332240243365083923142381781, −3.59311585959193498456883699520, −2.72438617014065795337000930010, −1.73086583988069406829045991620, −0.809933861033691545783734731021,
0.809933861033691545783734731021, 1.73086583988069406829045991620, 2.72438617014065795337000930010, 3.59311585959193498456883699520, 4.37332240243365083923142381781, 5.06645566925151233570382408451, 5.78867423767551983272336508703, 6.74170906319219068322168674229, 7.02018422312595018715997814354, 8.119344141437620494656346966862