| L(s) = 1 | + 2·2-s + 2·4-s + (2 + i)5-s + 3·7-s + (4 + 2i)10-s − 5i·11-s + (−3 + 2i)13-s + 6·14-s − 4·16-s + 3i·17-s + 6i·19-s + (4 + 2i)20-s − 10i·22-s − 9i·23-s + (3 + 4i)25-s + (−6 + 4i)26-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + (0.894 + 0.447i)5-s + 1.13·7-s + (1.26 + 0.632i)10-s − 1.50i·11-s + (−0.832 + 0.554i)13-s + 1.60·14-s − 16-s + 0.727i·17-s + 1.37i·19-s + (0.894 + 0.447i)20-s − 2.13i·22-s − 1.87i·23-s + (0.600 + 0.800i)25-s + (−1.17 + 0.784i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.49680 + 0.217703i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.49680 + 0.217703i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 13 | \( 1 + (3 - 2i)T \) |
| good | 2 | \( 1 - 2T + 2T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 5iT - 11T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 5iT - 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 5iT - 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + iT - 89T^{2} \) |
| 97 | \( 1 - 3T + 97T^{2} \) |
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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
−10.93920089835723928221966530646, −10.14089860039039578414241450246, −8.856831722026921941714036657255, −8.045242613288293194955200531335, −6.61900283515020463328930676205, −5.98287322860284498591810825816, −5.16056740807084470282749955218, −4.21242420726856230888366215458, −3.02844831692419744500769815553, −1.90306447948722643718359930552,
1.81519697457318625967382379853, 2.85497385576384006808620357979, 4.59390406039760231689655375873, 4.88271407134164340871263453969, 5.66842890621948180439326955837, 6.94627373863877983068362298647, 7.72419418467460300796419528806, 9.194915851832507433526147382543, 9.704887484850753901636210390500, 10.97844215882408177988573474519
