L(s) = 1 | − 3-s + (2 − i)5-s + i·7-s + 9-s + 4i·11-s − 2·13-s + (−2 + i)15-s + 2i·17-s − 8i·19-s − i·21-s − 4i·23-s + (3 − 4i)25-s − 27-s − 2i·29-s + 4·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + (0.894 − 0.447i)5-s + 0.377i·7-s + 0.333·9-s + 1.20i·11-s − 0.554·13-s + (−0.516 + 0.258i)15-s + 0.485i·17-s − 1.83i·19-s − 0.218i·21-s − 0.834i·23-s + (0.600 − 0.800i)25-s − 0.192·27-s − 0.371i·29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.737996728\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.737996728\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 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
−8.678959843910684354535765281541, −7.85075297509681791674816401126, −6.72349154604933063631531498362, −6.54242646286788321098726106374, −5.38097702192995466349952942588, −4.89154385998927961098074804378, −4.24496581814966021763416399108, −2.63962210547780052340180394054, −2.05876307779322191926962129719, −0.74138158525048299141490105584,
0.901160026147059571944260747147, 1.99499261803867860037445193670, 3.13365916906157826763865428526, 3.89417464597401757342466764224, 5.12769437408087778877743736325, 5.65372024085154137612866265216, 6.32519961467632881753222059219, 7.03169653556451437745169327320, 7.85736895862863114998902338165, 8.634417168796602573232200212981