L(s) = 1 | − 8i·5-s − 78.3i·7-s − 107.·11-s + 216i·13-s + 162·17-s − 440.·19-s − 705. i·23-s + 561·25-s − 1.30e3i·29-s + 627. i·31-s − 627.·35-s + 1.51e3i·37-s − 1.89e3·41-s − 2.90e3·43-s + 1.41e3i·47-s + ⋯ |
L(s) = 1 | − 0.320i·5-s − 1.59i·7-s − 0.890·11-s + 1.27i·13-s + 0.560·17-s − 1.22·19-s − 1.33i·23-s + 0.897·25-s − 1.55i·29-s + 0.652i·31-s − 0.511·35-s + 1.10i·37-s − 1.12·41-s − 1.57·43-s + 0.638i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5178802432\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5178802432\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 8iT - 625T^{2} \) |
| 7 | \( 1 + 78.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 107.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 216iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 162T + 8.35e4T^{2} \) |
| 19 | \( 1 + 440.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 705. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.30e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 627. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.51e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.89e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.90e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.41e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.97e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 2.26e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 2.37e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 1.67e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 7.75e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.75e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 7.99e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.33e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 2.43e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 7.45e3T + 8.85e7T^{2} \) |
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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
−9.561343807405751036548565674886, −8.458883441224840682300163725368, −7.910852334402969508343094696686, −6.83435865464586927037342008683, −6.39674219921815140614559443301, −4.82098754982617094350144880767, −4.45249123042718133300085656995, −3.36485861475174179310827934315, −2.06104512877774804684490008831, −0.899375656574012555344506807654,
0.12113231724545790396343909530, 1.78213705990231868549985650750, 2.76590957787220486317759309251, 3.45717730873494792467102620800, 5.19224436626559963113623584177, 5.41024345491123009443467511594, 6.43014329700367344424045704164, 7.48576432565562526562510198249, 8.341459391786133969637377743354, 8.873352500699933731476441658564