| L(s) = 1 | + (2.91 − 0.707i)3-s + 5.83·7-s + (8 − 4.12i)9-s + 16.4i·11-s − 11.3i·17-s + 12·19-s + (17 − 4.12i)21-s + 24.0i·23-s + (20.4 − 17.6i)27-s + 32·31-s + (11.6 + 48.0i)33-s − 23.3·37-s + 57.7i·41-s + 40.8·43-s + 35.3i·47-s + ⋯ |
| L(s) = 1 | + (0.971 − 0.235i)3-s + 0.832·7-s + (0.888 − 0.458i)9-s + 1.49i·11-s − 0.665i·17-s + 0.631·19-s + (0.809 − 0.196i)21-s + 1.04i·23-s + (0.755 − 0.654i)27-s + 1.03·31-s + (0.353 + 1.45i)33-s − 0.630·37-s + 1.40i·41-s + 0.949·43-s + 0.752i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.290183062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.290183062\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.91 + 0.707i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5.83T + 49T^{2} \) |
| 11 | \( 1 - 16.4iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 11.3iT - 289T^{2} \) |
| 19 | \( 1 - 12T + 361T^{2} \) |
| 23 | \( 1 - 24.0iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 32T + 961T^{2} \) |
| 37 | \( 1 + 23.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 57.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 67.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 16.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16T + 3.72e3T^{2} \) |
| 67 | \( 1 + 5.83T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 116.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 72T + 6.24e3T^{2} \) |
| 83 | \( 1 + 43.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 65.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 163.T + 9.40e3T^{2} \) |
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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
−9.642116467732150666327059692543, −8.757962352551235345529043417405, −7.76861706763498426903012439311, −7.44808330435580597415645397282, −6.48581659268062826958060133545, −5.06350918940668859887237603715, −4.44690160496100794466031510392, −3.27034247340394124367875606633, −2.19290996970596016459492930421, −1.28808868642069214270100484425,
1.00292540316474624783634881548, 2.28563088645017281250323724353, 3.29980427480520679415704628265, 4.17917065910469650127690282118, 5.16765947784305652682658031117, 6.16109198759364986931253941388, 7.26804307180761866137214330275, 8.191143999667589106444268097133, 8.540312570601821420223726674985, 9.329401043249144140480194845840
