L(s) = 1 | + (1.26 − 1.26i)3-s + (−1.98 + 1.75i)7-s − 0.201i·9-s + 4.28·11-s + (−4.40 + 4.40i)13-s + (3.77 + 3.77i)17-s − 8.01·19-s + (−0.292 + 4.72i)21-s + (2.07 + 2.07i)23-s + (3.54 + 3.54i)27-s − 0.383i·29-s + 1.01i·31-s + (5.41 − 5.41i)33-s + (−5.30 + 5.30i)37-s + 11.1i·39-s + ⋯ |
L(s) = 1 | + (0.730 − 0.730i)3-s + (−0.749 + 0.662i)7-s − 0.0670i·9-s + 1.29·11-s + (−1.22 + 1.22i)13-s + (0.914 + 0.914i)17-s − 1.83·19-s + (−0.0638 + 1.03i)21-s + (0.432 + 0.432i)23-s + (0.681 + 0.681i)27-s − 0.0712i·29-s + 0.182i·31-s + (0.942 − 0.942i)33-s + (−0.871 + 0.871i)37-s + 1.78i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.665075827\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.665075827\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.98 - 1.75i)T \) |
good | 3 | \( 1 + (-1.26 + 1.26i)T - 3iT^{2} \) |
| 11 | \( 1 - 4.28T + 11T^{2} \) |
| 13 | \( 1 + (4.40 - 4.40i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.77 - 3.77i)T + 17iT^{2} \) |
| 19 | \( 1 + 8.01T + 19T^{2} \) |
| 23 | \( 1 + (-2.07 - 2.07i)T + 23iT^{2} \) |
| 29 | \( 1 + 0.383iT - 29T^{2} \) |
| 31 | \( 1 - 1.01iT - 31T^{2} \) |
| 37 | \( 1 + (5.30 - 5.30i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.42iT - 41T^{2} \) |
| 43 | \( 1 + (-4.67 - 4.67i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.51 + 6.51i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.18 - 6.18i)T + 53iT^{2} \) |
| 59 | \( 1 - 9.39T + 59T^{2} \) |
| 61 | \( 1 + 1.99iT - 61T^{2} \) |
| 67 | \( 1 + (0.224 - 0.224i)T - 67iT^{2} \) |
| 71 | \( 1 - 7.88T + 71T^{2} \) |
| 73 | \( 1 + (9.62 - 9.62i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.59iT - 79T^{2} \) |
| 83 | \( 1 + (-6.32 + 6.32i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.04T + 89T^{2} \) |
| 97 | \( 1 + (4.17 + 4.17i)T + 97iT^{2} \) |
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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
−9.530526857631864397124541314259, −8.769745164079012185614452864686, −8.294788745739872128192579713525, −7.02332956304427050716322556751, −6.75263567657426663375155120934, −5.74227460531425578557038949099, −4.48751473685472607628827807886, −3.52278690791825744981910136531, −2.38354618937910989805320870283, −1.63645951083893913503893430650,
0.60498498775213869542892464260, 2.50599188634054749476705836606, 3.41519824582431015974148535219, 4.09215620285460308005119042591, 5.03730952656165007025943257925, 6.26810190739128832366458623236, 7.00227878886569759948652439370, 7.87326461123876198395087349835, 8.873546855445052875860480395630, 9.412303082795844974226951341180