L(s) = 1 | − i·9-s + (−1 − i)13-s + (−1 − i)17-s − 2·29-s + (−1 + i)37-s − i·49-s + (1 + i)53-s − 2i·61-s + (1 − i)73-s − 81-s + (1 + i)97-s + (1 − i)113-s + (−1 + i)117-s + ⋯ |
L(s) = 1 | − i·9-s + (−1 − i)13-s + (−1 − i)17-s − 2·29-s + (−1 + i)37-s − i·49-s + (1 + i)53-s − 2i·61-s + (1 − i)73-s − 81-s + (1 + i)97-s + (1 − i)113-s + (−1 + i)117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7357164886\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7357164886\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + (1 + i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + 2T + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1 - i)T + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + 2iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−8.708610726679551382837449125380, −7.74922393383044929538967402887, −7.11457656648307350791939560277, −6.43461904558354509306971422290, −5.46263949197854850011078557801, −4.85389042418997605937052989961, −3.78070882516106446624104476266, −3.01384498110321250829265305199, −1.99630123309310964539548613852, −0.40864308719317885893019546828,
1.84982270941990949294265551047, 2.34613659879259446539720216092, 3.75663845108611906175238596697, 4.44390618092104692110982104190, 5.27287145102835076675358530759, 6.02860693883933144435531006341, 7.10170024991443914354257803399, 7.41660139795826743974563637905, 8.446464346721647461212176906445, 9.018526991209342829402860701965