L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s − 7-s + (0.707 − 0.707i)8-s + 1.41·11-s + (0.866 − 0.5i)13-s + (−0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s + i·19-s + (1.36 − 0.366i)22-s + (−1.22 + 0.707i)23-s + (0.5 + 0.866i)25-s + (0.707 − 0.707i)26-s + (−0.866 + 0.499i)28-s + (−0.707 − 1.22i)29-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s − 7-s + (0.707 − 0.707i)8-s + 1.41·11-s + (0.866 − 0.5i)13-s + (−0.965 + 0.258i)14-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s + i·19-s + (1.36 − 0.366i)22-s + (−1.22 + 0.707i)23-s + (0.5 + 0.866i)25-s + (0.707 − 0.707i)26-s + (−0.866 + 0.499i)28-s + (−0.707 − 1.22i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.897892527\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.897892527\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + iT - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.664540665487213913443580899123, −9.217321508354016527721705638601, −7.926428704350266602507879608877, −6.95422267772213781506828811564, −6.17262892353418037970524004743, −5.77271881217281302862697773064, −4.28778919219920853946433880112, −3.78394146839429030441905537050, −2.79484770197549617263948078139, −1.45157312927322497613592631426,
1.81321724579706628788823073695, 3.06827917098239451321384913778, 3.99360414659820788064266153440, 4.57289299100515247870989876542, 5.97132362634341421819158647402, 6.68458334463651323814165361439, 6.77100333495075315837130366968, 8.386934381627424615196995183814, 8.855028546787588983880095380898, 9.921007298064378938316095999726