L(s) = 1 | + (−0.587 − 0.809i)4-s + (−0.809 + 0.587i)5-s + (0.309 + 0.0489i)7-s + (−0.951 + 0.309i)9-s + (−0.587 + 0.809i)11-s + (−0.309 + 0.951i)16-s + (−1.58 + 0.809i)17-s + (0.809 + 0.587i)19-s + (0.951 + 0.309i)20-s + (0.642 + 0.642i)23-s + (0.309 − 0.951i)25-s + (−0.142 − 0.278i)28-s + (−0.278 + 0.142i)35-s + (0.809 + 0.587i)36-s + (−0.642 + 0.642i)43-s + 44-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)4-s + (−0.809 + 0.587i)5-s + (0.309 + 0.0489i)7-s + (−0.951 + 0.309i)9-s + (−0.587 + 0.809i)11-s + (−0.309 + 0.951i)16-s + (−1.58 + 0.809i)17-s + (0.809 + 0.587i)19-s + (0.951 + 0.309i)20-s + (0.642 + 0.642i)23-s + (0.309 − 0.951i)25-s + (−0.142 − 0.278i)28-s + (−0.278 + 0.142i)35-s + (0.809 + 0.587i)36-s + (−0.642 + 0.642i)43-s + 44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4004524014\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4004524014\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (0.587 - 0.809i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
good | 2 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 3 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.0489i)T + (0.951 + 0.309i)T^{2} \) |
| 13 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (1.58 - 0.809i)T + (0.587 - 0.809i)T^{2} \) |
| 23 | \( 1 + (-0.642 - 0.642i)T + iT^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (0.642 - 0.642i)T - iT^{2} \) |
| 47 | \( 1 + (1.76 - 0.278i)T + (0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.221 - 1.39i)T + (-0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
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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
−10.40845825294976579252552074640, −9.651433540513964990730755844117, −8.608584064226203483188459376608, −8.054265945123752721775581879364, −7.03026979979695134316884379693, −6.10236081491530764120460602541, −5.10593604886094766885098307577, −4.41031743435027963341026339995, −3.19737034919313375151934384878, −1.88880212117456364992933003052,
0.36420530905654005016788826966, 2.75249697126039426628457124516, 3.52603094134415781395965133426, 4.72342655190488799935972499065, 5.17363227975242912777876438370, 6.63440647439017926791671910131, 7.56618074844156198356287470855, 8.449073090173689457273203829376, 8.728931942972028265829193078643, 9.536764231787602299011499946588