L(s) = 1 | + (−1 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s − 2.44·6-s + (−1.87 + 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s + 4.18·11-s + (2.44 − 2.44i)12-s + (−12.9 − 12.9i)13-s − 3.74i·14-s − 4·16-s + (23.0 − 23.0i)17-s + (−2.99 − 2.99i)18-s + 32.6i·19-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s − 0.408·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + 0.380·11-s + (0.204 − 0.204i)12-s + (−0.999 − 0.999i)13-s − 0.267i·14-s − 0.250·16-s + (1.35 − 1.35i)17-s + (−0.166 − 0.166i)18-s + 1.71i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.518446305\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.518446305\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
good | 11 | \( 1 - 4.18T + 121T^{2} \) |
| 13 | \( 1 + (12.9 + 12.9i)T + 169iT^{2} \) |
| 17 | \( 1 + (-23.0 + 23.0i)T - 289iT^{2} \) |
| 19 | \( 1 - 32.6iT - 361T^{2} \) |
| 23 | \( 1 + (-13.5 - 13.5i)T + 529iT^{2} \) |
| 29 | \( 1 - 23.2iT - 841T^{2} \) |
| 31 | \( 1 - 32.2T + 961T^{2} \) |
| 37 | \( 1 + (-21.0 + 21.0i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 51.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-24.7 - 24.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (52.0 - 52.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (26.3 + 26.3i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 103. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 119.T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-49.4 + 49.4i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 94.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-15.0 - 15.0i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 112. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (13.6 + 13.6i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 35.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (72.1 - 72.1i)T - 9.40e3iT^{2} \) |
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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
−9.823206503283132915701197248039, −9.277113336323105610228512047474, −8.110234848385367855970145984745, −7.71396575436347263927055721892, −6.71908174879233681408713450377, −5.55500196125029209464640519757, −5.03089844130532868487980361779, −3.58029450445980521398017193279, −2.68624666211677724334210024954, −1.08370642359875662801880795558,
0.62905877907923150572258735944, 1.90092172207297677750823483593, 2.91287027929317081897136838536, 3.95408415441468099800470654092, 4.97910151505228025876763398364, 6.51698267560434873313616204110, 6.97465456093044616987767608976, 8.030076963560764700602040402391, 8.659040219254798113627808446997, 9.655494491933572613215856442820