L(s) = 1 | + (−0.588 − 2.19i)3-s + (1.03 + 1.98i)5-s + (−2.24 + 1.40i)7-s + (−1.88 + 1.08i)9-s + (−1.78 + 3.09i)11-s + (−3.13 + 3.13i)13-s + (3.74 − 3.44i)15-s + (−5.00 + 1.34i)17-s + (−0.687 − 1.19i)19-s + (4.40 + 4.10i)21-s + (−1.07 + 4.00i)23-s + (−2.85 + 4.10i)25-s + (−1.32 − 1.32i)27-s − 9.39i·29-s + (6.08 + 3.51i)31-s + ⋯ |
L(s) = 1 | + (−0.340 − 1.26i)3-s + (0.462 + 0.886i)5-s + (−0.847 + 0.530i)7-s + (−0.628 + 0.363i)9-s + (−0.539 + 0.934i)11-s + (−0.869 + 0.869i)13-s + (0.967 − 0.888i)15-s + (−1.21 + 0.325i)17-s + (−0.157 − 0.273i)19-s + (0.961 + 0.895i)21-s + (−0.223 + 0.835i)23-s + (−0.571 + 0.820i)25-s + (−0.254 − 0.254i)27-s − 1.74i·29-s + (1.09 + 0.630i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0902 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0902 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.428776 + 0.469382i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.428776 + 0.469382i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.03 - 1.98i)T \) |
| 7 | \( 1 + (2.24 - 1.40i)T \) |
good | 3 | \( 1 + (0.588 + 2.19i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (1.78 - 3.09i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.13 - 3.13i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.00 - 1.34i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.687 + 1.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.07 - 4.00i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 9.39iT - 29T^{2} \) |
| 31 | \( 1 + (-6.08 - 3.51i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.29 - 1.68i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 4.63iT - 41T^{2} \) |
| 43 | \( 1 + (2.51 + 2.51i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.34 - 8.76i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.75 + 0.470i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (1.42 - 2.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.91 + 1.10i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.17 + 4.37i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + (-0.654 - 2.44i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (4.82 - 2.78i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.863 + 0.863i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.430 + 0.745i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (12.6 + 12.6i)T + 97iT^{2} \) |
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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
−11.22535477868008627557869011658, −9.905225552002971609780787976889, −9.539224907791142494637189585468, −8.068916540595434253462610461680, −7.10575037917070483370460103211, −6.59675542916172461610673123490, −5.90984036664898808460680729303, −4.45911995137189246316562975201, −2.66006833854484945610607861114, −2.00971983208635492607771347835,
0.35034002967831843917986264740, 2.73555405550229607480103457519, 3.99971191595738446700833446933, 4.90168329371094626402249864883, 5.63242946879887139944597525441, 6.71888418062002193252406447204, 8.113595465031328285659459732886, 9.006770924152987583619033685947, 9.803611720472626628433935167250, 10.39273274104997684273642805394