L(s) = 1 | + (−0.923 + 0.382i)3-s + (−1.30 − 1.30i)7-s + (0.707 − 0.707i)9-s + (−0.292 − 0.707i)13-s + (0.541 + 1.30i)19-s + (1.70 + 0.707i)21-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s − 0.765·31-s + (−0.707 + 1.70i)37-s + (0.541 + 0.541i)39-s + (−1.30 − 0.541i)43-s + 2.41i·49-s + (−1 − 0.999i)57-s + (−0.707 + 0.292i)61-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.382i)3-s + (−1.30 − 1.30i)7-s + (0.707 − 0.707i)9-s + (−0.292 − 0.707i)13-s + (0.541 + 1.30i)19-s + (1.70 + 0.707i)21-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s − 0.765·31-s + (−0.707 + 1.70i)37-s + (0.541 + 0.541i)39-s + (−1.30 − 0.541i)43-s + 2.41i·49-s + (−1 − 0.999i)57-s + (−0.707 + 0.292i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02615694568\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02615694568\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.923 - 0.382i)T \) |
good | 5 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + 0.765T + T^{2} \) |
| 37 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 79 | \( 1 + 0.765iT - T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−8.461449994943677177270621358222, −7.51283165889050860158582563591, −6.92509804742919480906110204176, −6.19065234587452365386363370962, −5.54556081253538168218500376455, −4.55613874738925135628658178589, −3.72983181988618739584630407791, −3.17264193500663796448231238244, −1.36325965665120320497667601185, −0.01838587961607246762486038498,
1.81116589210094898796178619029, 2.72570548388091537516946703899, 3.76201767645481751916971124095, 4.96852469813079910728370548389, 5.54648854803322094515392074315, 6.26417886995819645838024395513, 6.91255583449304109759819518682, 7.51515405356025542165830044433, 8.733489879752992615724083852520, 9.357012395680894944436117102214