| L(s) = 1 | + i·2-s − 4-s − 2i·7-s − i·8-s + i·13-s + 2·14-s + 16-s − 2·19-s + 6i·23-s − 26-s + 2i·28-s − 4·31-s + i·32-s − 2i·37-s − 2i·38-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.755i·7-s − 0.353i·8-s + 0.277i·13-s + 0.534·14-s + 0.250·16-s − 0.458·19-s + 1.25i·23-s − 0.196·26-s + 0.377i·28-s − 0.718·31-s + 0.176i·32-s − 0.328i·37-s − 0.324i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.262186951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.262186951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.169385420287446907991853890329, −7.43689563544388306928317870698, −7.12212854074895949071130231258, −6.18481797254258035466347439644, −5.61231806324589921026656971107, −4.72236309063112340841944306961, −4.03074970426613406889338523094, −3.34602889051150886701336101414, −2.07908366386907650415466676821, −0.946340358144225596878288945148,
0.38349345436385057722912645966, 1.68215248756078875218895725144, 2.51705140556881454726675404655, 3.19151900049041085647312607639, 4.20372234779582980550909298256, 4.83002052795350724146455741050, 5.71960972394324912791380426165, 6.27983602961984556122515326997, 7.24219985745763752113086955983, 8.059826631698065938273243112727
