L(s) = 1 | − i·2-s − 2i·3-s + 4-s + (2 − i)5-s − 2·6-s − 3i·8-s − 9-s + (−1 − 2i)10-s − 2·11-s − 2i·12-s + (−2 − 4i)15-s − 16-s + i·18-s − 6·19-s + (2 − i)20-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.15i·3-s + 0.5·4-s + (0.894 − 0.447i)5-s − 0.816·6-s − 1.06i·8-s − 0.333·9-s + (−0.316 − 0.632i)10-s − 0.603·11-s − 0.577i·12-s + (−0.516 − 1.03i)15-s − 0.250·16-s + 0.235i·18-s − 1.37·19-s + (0.447 − 0.223i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.475780 - 2.01543i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.475780 - 2.01543i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2 + i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−10.14369918590582487730351557528, −9.039565531077304240188068414120, −8.087711438496173037540703391661, −7.15744808508653996101297506864, −6.42642892346781123023563355893, −5.65560871970331382424842091566, −4.29337848099055378161232012987, −2.73947032183209890753205430055, −2.00485295887563018000026616432, −1.02764169643058473740593148586,
2.13076582645055884157808769733, 3.10073755491167276543285795263, 4.58291719282647597348243608599, 5.24770428263861635359632695637, 6.39615200058717896816035661040, 6.74978280820196409084840839384, 8.142116494652709980814741667129, 8.755498079935120604911876661826, 10.01910007183442033340494949047, 10.35134645824883003215451195087