L(s) = 1 | − 2·4-s − 5-s + 7-s + 2·11-s − 5·13-s + 4·16-s + 4·19-s + 2·20-s + 5·23-s + 25-s − 2·28-s − 6·29-s + 3·31-s − 35-s + 7·37-s + 3·41-s + 6·43-s − 4·44-s − 47-s + 49-s + 10·52-s − 4·53-s − 2·55-s + 6·59-s − 61-s − 8·64-s + 5·65-s + ⋯ |
L(s) = 1 | − 4-s − 0.447·5-s + 0.377·7-s + 0.603·11-s − 1.38·13-s + 16-s + 0.917·19-s + 0.447·20-s + 1.04·23-s + 1/5·25-s − 0.377·28-s − 1.11·29-s + 0.538·31-s − 0.169·35-s + 1.15·37-s + 0.468·41-s + 0.914·43-s − 0.603·44-s − 0.145·47-s + 1/7·49-s + 1.38·52-s − 0.549·53-s − 0.269·55-s + 0.781·59-s − 0.128·61-s − 64-s + 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.25574541042607, −13.55224749081981, −13.19149870049226, −12.50026361869632, −12.29909708606554, −11.65972543571579, −11.17187635779562, −10.73542800895187, −9.891455214911388, −9.521829180464489, −9.294688627622472, −8.612780121324914, −8.011675486867025, −7.624892500744765, −7.152612990052108, −6.530172682252109, −5.715961592998442, −5.157353629991572, −4.893646641897034, −4.058135698446168, −3.902732384890995, −2.934553651906595, −2.501101045564630, −1.396677005960422, −0.8539212809883762, 0,
0.8539212809883762, 1.396677005960422, 2.501101045564630, 2.934553651906595, 3.902732384890995, 4.058135698446168, 4.893646641897034, 5.157353629991572, 5.715961592998442, 6.530172682252109, 7.152612990052108, 7.624892500744765, 8.011675486867025, 8.612780121324914, 9.294688627622472, 9.521829180464489, 9.891455214911388, 10.73542800895187, 11.17187635779562, 11.65972543571579, 12.29909708606554, 12.50026361869632, 13.19149870049226, 13.55224749081981, 14.25574541042607