L(s) = 1 | + (−2.09 + 2.09i)3-s + (2.59 + 0.510i)7-s − 5.75i·9-s − 3.94·11-s + (1.69 − 1.69i)13-s + (2.66 + 2.66i)17-s + 5.36·19-s + (−6.50 + 4.36i)21-s + (3.55 + 3.55i)23-s + (5.77 + 5.77i)27-s − 7.24i·29-s − 0.174i·31-s + (8.26 − 8.26i)33-s + (4.85 − 4.85i)37-s + 7.10i·39-s + ⋯ |
L(s) = 1 | + (−1.20 + 1.20i)3-s + (0.981 + 0.193i)7-s − 1.91i·9-s − 1.19·11-s + (0.470 − 0.470i)13-s + (0.647 + 0.647i)17-s + 1.22·19-s + (−1.41 + 0.952i)21-s + (0.741 + 0.741i)23-s + (1.11 + 1.11i)27-s − 1.34i·29-s − 0.0312i·31-s + (1.43 − 1.43i)33-s + (0.798 − 0.798i)37-s + 1.13i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0631 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0631 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.169814103\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.169814103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.59 - 0.510i)T \) |
good | 3 | \( 1 + (2.09 - 2.09i)T - 3iT^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 13 | \( 1 + (-1.69 + 1.69i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.66 - 2.66i)T + 17iT^{2} \) |
| 19 | \( 1 - 5.36T + 19T^{2} \) |
| 23 | \( 1 + (-3.55 - 3.55i)T + 23iT^{2} \) |
| 29 | \( 1 + 7.24iT - 29T^{2} \) |
| 31 | \( 1 + 0.174iT - 31T^{2} \) |
| 37 | \( 1 + (-4.85 + 4.85i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.732iT - 41T^{2} \) |
| 43 | \( 1 + (-8.20 - 8.20i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.31 - 2.31i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.53 + 4.53i)T + 53iT^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 13.3iT - 61T^{2} \) |
| 67 | \( 1 + (6.55 - 6.55i)T - 67iT^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + (7.02 - 7.02i)T - 73iT^{2} \) |
| 79 | \( 1 + 6.63iT - 79T^{2} \) |
| 83 | \( 1 + (10.4 - 10.4i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.43T + 89T^{2} \) |
| 97 | \( 1 + (1.40 + 1.40i)T + 97iT^{2} \) |
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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
−9.901371765481420551616159267043, −9.229261104248480582279680872798, −8.048995114082528211203420255040, −7.49702112705716742737117686160, −5.88923954846910948620471319453, −5.63708466008710730895304966944, −4.83675522690126408788490142224, −4.02442525316880764963233492524, −2.85965759622582067216906392735, −1.02833751853881732165471022309,
0.74408642969350237436735617670, 1.70412576497788478605122165695, 2.99303520570132795884511417554, 4.75290366008183342933394313345, 5.23064927542848185622696185959, 6.03301483689573628903900147528, 7.07516337161270755943973159113, 7.53928934982824703375191035115, 8.234500932607730446544732321643, 9.350818614323398030424060002609