| L(s) = 1 | + 7-s + 3·11-s − 13-s − 3·17-s + 7·19-s − 9·23-s + 9·29-s + 4·31-s + 7·37-s − 12·41-s − 43-s + 49-s − 6·53-s + 12·59-s − 61-s + 14·67-s + 12·71-s + 7·73-s + 3·77-s + 10·79-s + 6·83-s + 6·89-s − 91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.904·11-s − 0.277·13-s − 0.727·17-s + 1.60·19-s − 1.87·23-s + 1.67·29-s + 0.718·31-s + 1.15·37-s − 1.87·41-s − 0.152·43-s + 1/7·49-s − 0.824·53-s + 1.56·59-s − 0.128·61-s + 1.71·67-s + 1.42·71-s + 0.819·73-s + 0.341·77-s + 1.12·79-s + 0.658·83-s + 0.635·89-s − 0.104·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.492825035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.492825035\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.32511515229606, −12.07893573128784, −11.83219207315013, −11.31352956929444, −10.90066484943771, −10.15741903345667, −9.858664864076558, −9.556084274510375, −8.955304991713839, −8.328815273036390, −8.062045499556725, −7.683651402880347, −6.807557442789777, −6.627375779462712, −6.216206010914783, −5.449360691331366, −5.021244984805128, −4.607001017552949, −3.895716217863216, −3.641123559866578, −2.863688157322583, −2.301199303275998, −1.773198530827646, −1.042429774076317, −0.5539770868698499,
0.5539770868698499, 1.042429774076317, 1.773198530827646, 2.301199303275998, 2.863688157322583, 3.641123559866578, 3.895716217863216, 4.607001017552949, 5.021244984805128, 5.449360691331366, 6.216206010914783, 6.627375779462712, 6.807557442789777, 7.683651402880347, 8.062045499556725, 8.328815273036390, 8.955304991713839, 9.556084274510375, 9.858664864076558, 10.15741903345667, 10.90066484943771, 11.31352956929444, 11.83219207315013, 12.07893573128784, 12.32511515229606
