| L(s) = 1 | + 2-s + 4-s − 1.41i·7-s + 8-s + (0.772 + 3.22i)11-s − 3.62i·13-s − 1.41i·14-s + 16-s + 2·17-s + 6.07i·19-s + (0.772 + 3.22i)22-s − 7.77i·23-s − 3.62i·26-s − 1.41i·28-s + 5.49·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.534i·7-s + 0.353·8-s + (0.232 + 0.972i)11-s − 1.00i·13-s − 0.377i·14-s + 0.250·16-s + 0.485·17-s + 1.39i·19-s + (0.164 + 0.687i)22-s − 1.62i·23-s − 0.710i·26-s − 0.267i·28-s + 1.02·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.313403362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.313403362\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-0.772 - 3.22i)T \) |
| good | 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 13 | \( 1 + 3.62iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 6.07iT - 19T^{2} \) |
| 23 | \( 1 + 7.77iT - 23T^{2} \) |
| 29 | \( 1 - 5.49T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 1.54T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 + 9.45iT - 43T^{2} \) |
| 47 | \( 1 - 9.96iT - 47T^{2} \) |
| 53 | \( 1 - 6.07iT - 53T^{2} \) |
| 59 | \( 1 - 0.794iT - 59T^{2} \) |
| 61 | \( 1 + 9.96iT - 61T^{2} \) |
| 67 | \( 1 - 3.08T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 4.06iT - 73T^{2} \) |
| 79 | \( 1 - 6.07iT - 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−8.021469767944944953990098499097, −7.49052382952952984159654428545, −6.59441446705715271027879087016, −6.11935311507783778820368743746, −5.11428031807755091445146407660, −4.55134877784578728451722156133, −3.74985633042571856303393436679, −2.95748210474547185666543547584, −1.97787410671596045402707066953, −0.842873623447919928117733094511,
1.01478202076891369695974352282, 2.15461796167232504901795469830, 3.06490842143612243763562040428, 3.70588684089231866277148613205, 4.74704988528178955615500446738, 5.25328560388443044139391401832, 6.16356203927658669177897774182, 6.65413221489143132815940144108, 7.42502113308543634169374852426, 8.373418652336487452802326409589
