L(s) = 1 | + (−1.41 + 1.73i)5-s + 2.82i·7-s + 4.89·11-s + 4.89i·13-s − 3.46i·17-s − 6.92·19-s + 4i·23-s + (−0.999 − 4.89i)25-s + 8.48·29-s − 6.92·31-s + (−4.89 − 4.00i)35-s − 4.89i·37-s − 5.65·41-s + 11.3i·43-s − 4i·47-s + ⋯ |
L(s) = 1 | + (−0.632 + 0.774i)5-s + 1.06i·7-s + 1.47·11-s + 1.35i·13-s − 0.840i·17-s − 1.58·19-s + 0.834i·23-s + (−0.199 − 0.979i)25-s + 1.57·29-s − 1.24·31-s + (−0.828 − 0.676i)35-s − 0.805i·37-s − 0.883·41-s + 1.72i·43-s − 0.583i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.107821686\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.107821686\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.41 - 1.73i)T \) |
good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 4.89iT - 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 11.3iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 9.79iT - 97T^{2} \) |
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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
−9.641339085714527650718594588231, −9.021746153601346965746882707771, −8.422071394968896028202443906302, −7.22453818464379769485479135870, −6.62741699744747468820819692294, −5.97714613215373633506788005513, −4.61128838382999255603943117983, −3.91228261139030945856362857482, −2.80328547497321388664078144788, −1.74663735586104670296101972234,
0.45550072171400632815060101520, 1.58864252528320257570338011938, 3.36354851816139648056029198162, 4.12029107992080823241793755540, 4.74854484650204742503646953158, 6.03386449849410135727880506344, 6.78476663878454424009094705540, 7.69201763488134023367250224808, 8.524831055888802075187955272815, 8.918966741923406764242557924015