| L(s) = 1 | − i·2-s − 4-s − 0.646i·5-s + (2.44 − i)7-s + i·8-s − 0.646·10-s + (−2.79 + 1.79i)11-s − 3.09·13-s + (−1 − 2.44i)14-s + 16-s − 3.74·17-s − 5.54·19-s + 0.646i·20-s + (1.79 + 2.79i)22-s − 4·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.288i·5-s + (0.925 − 0.377i)7-s + 0.353i·8-s − 0.204·10-s + (−0.841 + 0.540i)11-s − 0.858·13-s + (−0.267 − 0.654i)14-s + 0.250·16-s − 0.907·17-s − 1.27·19-s + 0.144i·20-s + (0.381 + 0.595i)22-s − 0.834·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3484802064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3484802064\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.44 + i)T \) |
| 11 | \( 1 + (2.79 - 1.79i)T \) |
| good | 5 | \( 1 + 0.646iT - 5T^{2} \) |
| 13 | \( 1 + 3.09T + 13T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7.58iT - 29T^{2} \) |
| 31 | \( 1 - 1.15iT - 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 + 11.1iT - 43T^{2} \) |
| 47 | \( 1 - 5.03iT - 47T^{2} \) |
| 53 | \( 1 - 2.41T + 53T^{2} \) |
| 59 | \( 1 - 3.09iT - 59T^{2} \) |
| 61 | \( 1 + 9.28T + 61T^{2} \) |
| 67 | \( 1 - 1.58T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 6.32T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 9.15T + 83T^{2} \) |
| 89 | \( 1 - 9.79iT - 89T^{2} \) |
| 97 | \( 1 - 15.9iT - 97T^{2} \) |
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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
−9.078198593159204957625293071808, −8.388997575106378603937073990968, −7.65544020786271130556750879237, −6.73993079734012989180684356485, −5.43889134487594145638606215481, −4.65228969717493853598518622416, −4.10009880352403347777561178965, −2.53761138395734820327141860571, −1.83718892309898971538699177901, −0.13063235393916447917351413837,
1.92430273093018105522092504308, 3.03684669497185071045077912079, 4.46816497701266335204399505408, 5.03468510411286841875783342983, 5.96690444497082575979600804959, 6.84472729598756716024855387930, 7.62480072870278002279339497565, 8.500714154532331950852086135198, 8.844625311322821311704352027082, 10.10975072242941527611152781429
