L(s) = 1 | − 1.90·3-s − 3.95i·7-s − 5.35·9-s − 6.18·11-s + 4.94i·13-s − 22.4·17-s + 10.3·19-s + 7.54i·21-s + 39.6i·23-s + 27.4·27-s − 30.4i·29-s + 24.7i·31-s + 11.8·33-s − 24.0i·37-s − 9.43i·39-s + ⋯ |
L(s) = 1 | − 0.636·3-s − 0.565i·7-s − 0.595·9-s − 0.562·11-s + 0.380i·13-s − 1.31·17-s + 0.546·19-s + 0.359i·21-s + 1.72i·23-s + 1.01·27-s − 1.04i·29-s + 0.797i·31-s + 0.357·33-s − 0.649i·37-s − 0.241i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9964431812\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9964431812\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 1.90T + 9T^{2} \) |
| 7 | \( 1 + 3.95iT - 49T^{2} \) |
| 11 | \( 1 + 6.18T + 121T^{2} \) |
| 13 | \( 1 - 4.94iT - 169T^{2} \) |
| 17 | \( 1 + 22.4T + 289T^{2} \) |
| 19 | \( 1 - 10.3T + 361T^{2} \) |
| 23 | \( 1 - 39.6iT - 529T^{2} \) |
| 29 | \( 1 + 30.4iT - 841T^{2} \) |
| 31 | \( 1 - 24.7iT - 961T^{2} \) |
| 37 | \( 1 + 24.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 31.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 52.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 13.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 17.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 104.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 57.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 99.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 16.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 139. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 91.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 94.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 143.T + 9.40e3T^{2} \) |
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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
−9.213219812044311212245460561482, −8.405110828348122837288003854590, −7.47528050371874687415260379616, −6.78641698816236036654585985636, −5.85526418105898228824434300312, −5.16344031915732219392128026573, −4.23541504063291322482517468473, −3.18866847845211668458651605028, −1.97826990523149098615027337746, −0.52631721043164102357902738478,
0.58912588050623727882392802603, 2.27010810291942887761746113225, 3.04713171498589523599481614812, 4.47108789831693061277673699578, 5.18310852647358627903186844687, 5.99755192559546434509427834445, 6.64883998395258342095788033675, 7.67466611095297407331573921103, 8.649196972022588069613002268382, 9.009709707025302911427449561374