| L(s) = 1 | + i·3-s − 3.46·7-s + 2·9-s − 3i·11-s + 3.46i·13-s + 3·17-s − i·19-s − 3.46i·21-s + 5i·27-s + 10.3i·29-s + 6.92·31-s + 3·33-s + 10.3i·37-s − 3.46·39-s − 9·41-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.30·7-s + 0.666·9-s − 0.904i·11-s + 0.960i·13-s + 0.727·17-s − 0.229i·19-s − 0.755i·21-s + 0.962i·27-s + 1.92i·29-s + 1.24·31-s + 0.522·33-s + 1.70i·37-s − 0.554·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.228190544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.228190544\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 11iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 15iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−9.828811446401796775896956406609, −8.947410594852934265868260043708, −8.256451753190954541397734939136, −6.85483436384713256707966737960, −6.68226693209946575585033866919, −5.49661617337831712320718593387, −4.60926622573821709970303468403, −3.56431351919034813383691231886, −3.02017442303665297041740427907, −1.31126170726735790853787954819,
0.50984259156794974918424587969, 1.95877629753816336500375151585, 3.06614619778292456830886088170, 3.99981201727396322075655547481, 5.09102860457684433509625386889, 6.20768381771529631096708504923, 6.65613428916529316904308832976, 7.65645677562659453668901912782, 8.076727273054881818807575581595, 9.469057445764487827895256104258
