| L(s) = 1 | + (1 − 1.41i)3-s + (−1.00 − 2.82i)9-s − 4.89·11-s − 4.89·13-s + 3.46i·17-s − 3.46i·19-s − 6·23-s + (−5.00 − 1.41i)27-s + 2.82i·29-s − 3.46i·31-s + (−4.89 + 6.92i)33-s − 4.89·37-s + (−4.89 + 6.92i)39-s − 5.65i·41-s − 8.48i·43-s + ⋯ |
| L(s) = 1 | + (0.577 − 0.816i)3-s + (−0.333 − 0.942i)9-s − 1.47·11-s − 1.35·13-s + 0.840i·17-s − 0.794i·19-s − 1.25·23-s + (−0.962 − 0.272i)27-s + 0.525i·29-s − 0.622i·31-s + (−0.852 + 1.20i)33-s − 0.805·37-s + (−0.784 + 1.10i)39-s − 0.883i·41-s − 1.29i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5787693741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5787693741\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 + 1.41i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8.48iT - 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.182774941869319016646864415937, −8.415916881480435391179893186400, −7.57483424343093841107880190691, −7.16926652991655292543140148327, −5.98556502377901711600667024005, −5.16861511076408231112749362784, −3.94536854376273629142937536651, −2.70018377210525305784776554293, −2.05428805373599319694578494794, −0.20404618580031993280575827241,
2.23837138895444839726466662636, 2.94307584961044825849348005237, 4.14946578398659871203693389002, 5.03309726173317248910347170342, 5.65237751979499147345805771279, 7.11577066413084289315137422344, 7.889500085471120137599591280857, 8.429160598143137717703044008262, 9.669905098205547748978338914904, 9.952738784947804271734731383611
