L(s) = 1 | − 4i·7-s − 4·11-s + 2i·13-s − 2i·17-s − 4·19-s + 4i·23-s − 2·29-s − 8·31-s + 6i·37-s + 6·41-s + 8i·43-s − 4i·47-s − 9·49-s + 6i·53-s − 4·59-s + ⋯ |
L(s) = 1 | − 1.51i·7-s − 1.20·11-s + 0.554i·13-s − 0.485i·17-s − 0.917·19-s + 0.834i·23-s − 0.371·29-s − 1.43·31-s + 0.986i·37-s + 0.937·41-s + 1.21i·43-s − 0.583i·47-s − 1.28·49-s + 0.824i·53-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.847709111336608870019396406632, −7.77080944611726142621936812250, −7.41172780847449720363020333335, −6.55968730212721699227884327880, −5.52542560750406947022530639259, −4.58885776614249694861552563657, −3.87672902355966487323261371011, −2.79112454019451606856737674244, −1.48499495784257744217639413273, 0,
2.06277990496082175008236103009, 2.67418249861115509209892080655, 3.85879349896762081795401640517, 5.09528650114999427237624895685, 5.62377263191614303710863542778, 6.36738877198213162555516852131, 7.52119316008319306292488441513, 8.243293411874429110793515009153, 8.873584566447424430706400357654