L(s) = 1 | + (1.36 − 0.382i)2-s − i·3-s + (1.70 − 1.04i)4-s + i·5-s + (−0.382 − 1.36i)6-s − 3.81·7-s + (1.92 − 2.07i)8-s − 9-s + (0.382 + 1.36i)10-s − 3.29·11-s + (−1.04 − 1.70i)12-s − 6.74·13-s + (−5.18 + 1.45i)14-s + 15-s + (1.83 − 3.55i)16-s + 2.05i·17-s + ⋯ |
L(s) = 1 | + (0.962 − 0.270i)2-s − 0.577i·3-s + (0.853 − 0.520i)4-s + 0.447i·5-s + (−0.156 − 0.555i)6-s − 1.44·7-s + (0.681 − 0.732i)8-s − 0.333·9-s + (0.120 + 0.430i)10-s − 0.992·11-s + (−0.300 − 0.492i)12-s − 1.87·13-s + (−1.38 + 0.389i)14-s + 0.258·15-s + (0.457 − 0.889i)16-s + 0.498i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3513982613\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3513982613\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 + 0.382i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-2.33 - 4.19i)T \) |
good | 7 | \( 1 + 3.81T + 7T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 13 | \( 1 + 6.74T + 13T^{2} \) |
| 17 | \( 1 - 2.05iT - 17T^{2} \) |
| 19 | \( 1 + 4.83T + 19T^{2} \) |
| 29 | \( 1 - 4.01T + 29T^{2} \) |
| 31 | \( 1 - 0.0146iT - 31T^{2} \) |
| 37 | \( 1 + 6.85iT - 37T^{2} \) |
| 41 | \( 1 + 4.62T + 41T^{2} \) |
| 43 | \( 1 - 4.10T + 43T^{2} \) |
| 47 | \( 1 + 4.94iT - 47T^{2} \) |
| 53 | \( 1 + 8.07iT - 53T^{2} \) |
| 59 | \( 1 + 7.38iT - 59T^{2} \) |
| 61 | \( 1 + 2.85iT - 61T^{2} \) |
| 67 | \( 1 - 0.919T + 67T^{2} \) |
| 71 | \( 1 - 5.96iT - 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 9.94T + 83T^{2} \) |
| 89 | \( 1 - 13.0iT - 89T^{2} \) |
| 97 | \( 1 + 4.51iT - 97T^{2} \) |
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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
−9.394987458892717211170245934585, −8.006236302969281137962402786420, −7.10921000429704731144447868882, −6.69766168847697379644498203166, −5.76895937933245473250711241789, −4.96983347868702347465239118680, −3.73204074440811854628334253046, −2.79650140502329232694676042442, −2.18850952949576554835016320583, −0.088818735096042755683040937562,
2.57631538119459847364143684375, 2.97535718851918848194670098352, 4.38365636945574369164932773629, 4.84521932703430707355349769765, 5.79276817716955227286869948864, 6.65939141731969829197123082027, 7.40191551532354296065160448034, 8.381615564400437867994774884089, 9.334222288256155686231485106123, 10.18526948685065638467421759372