L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 5·11-s − 15-s + 17-s − 6·19-s − 2·21-s − 3·23-s + 25-s − 27-s + 3·29-s + 7·31-s + 5·33-s + 2·35-s + 5·37-s + 2·41-s + 43-s + 45-s − 7·47-s − 3·49-s − 51-s + 6·53-s − 5·55-s + 6·57-s − 11·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 0.258·15-s + 0.242·17-s − 1.37·19-s − 0.436·21-s − 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 1.25·31-s + 0.870·33-s + 0.338·35-s + 0.821·37-s + 0.312·41-s + 0.152·43-s + 0.149·45-s − 1.02·47-s − 3/7·49-s − 0.140·51-s + 0.824·53-s − 0.674·55-s + 0.794·57-s − 1.43·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.79923540156747, −12.41430937471182, −11.73200242544846, −11.51923881625728, −10.86825335485148, −10.50145990732790, −10.23772777703043, −9.777320982340693, −9.186273855642017, −8.496093175948793, −8.180241189831203, −7.832370766032975, −7.302830821005061, −6.611232858573968, −6.238040888515778, −5.778038756436961, −5.290416861544722, −4.747759053728816, −4.474077549876928, −3.907161605576536, −2.915492547551112, −2.663608817972092, −1.971288225661149, −1.484354356358838, −0.6863412781763340, 0,
0.6863412781763340, 1.484354356358838, 1.971288225661149, 2.663608817972092, 2.915492547551112, 3.907161605576536, 4.474077549876928, 4.747759053728816, 5.290416861544722, 5.778038756436961, 6.238040888515778, 6.611232858573968, 7.302830821005061, 7.832370766032975, 8.180241189831203, 8.496093175948793, 9.186273855642017, 9.777320982340693, 10.23772777703043, 10.50145990732790, 10.86825335485148, 11.51923881625728, 11.73200242544846, 12.41430937471182, 12.79923540156747