L(s) = 1 | − 4.12i·2-s − 3·3-s − 9·4-s + 3.05i·5-s + 12.3i·6-s − 6.68i·7-s + 4.12i·8-s + 9·9-s + 12.5·10-s + 32.2i·11-s + 27·12-s − 27.5·14-s − 9.15i·15-s − 55·16-s + 28.8·17-s − 37.1i·18-s + ⋯ |
L(s) = 1 | − 1.45i·2-s − 0.577·3-s − 1.12·4-s + 0.272i·5-s + 0.841i·6-s − 0.360i·7-s + 0.182i·8-s + 0.333·9-s + 0.397·10-s + 0.883i·11-s + 0.649·12-s − 0.526·14-s − 0.157i·15-s − 0.859·16-s + 0.411·17-s − 0.485i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.519494969\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.519494969\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 4.12iT - 8T^{2} \) |
| 5 | \( 1 - 3.05iT - 125T^{2} \) |
| 7 | \( 1 + 6.68iT - 343T^{2} \) |
| 11 | \( 1 - 32.2iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 28.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 101. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 38.0iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 327. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 56.0iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 517. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 695.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 656. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 701.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 57.1iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 309. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 389. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 901.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 687. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.75e3iT - 9.12e5T^{2} \) |
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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
−10.42332569631937706071892921471, −9.849912040874679629049006654125, −8.808491673620562652901699306725, −7.41785088116631367900739766224, −6.60771376470093582973198682163, −5.21256761984273006692760158046, −4.20932840525056431537114674643, −3.20293690892617838063426145790, −1.91593430554675874171191260600, −0.77800311716528864538557325894,
0.828121668699752977646158261704, 2.92813762879801121316012322212, 4.71481401962390048473459176559, 5.24193588525661879466698730060, 6.31251880231694975893080498969, 6.84463824092277406030384069262, 8.005080914298824132518475147589, 8.702832904538426166680317201194, 9.544136386769674523357529948015, 10.90228126412125998073469600029