| L(s) = 1 | − 46.7i·3-s − 795.·5-s + 2.62e3i·7-s − 2.18e3·9-s + 1.00e4i·11-s + 1.10e4·13-s + 3.71e4i·15-s + 5.96e4·17-s − 1.35e5i·19-s + 1.22e5·21-s − 1.39e5i·23-s + 2.41e5·25-s + 1.02e5i·27-s + 6.21e5·29-s − 3.34e5i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.27·5-s + 1.09i·7-s − 0.333·9-s + 0.689i·11-s + 0.386·13-s + 0.734i·15-s + 0.713·17-s − 1.03i·19-s + 0.631·21-s − 0.499i·23-s + 0.618·25-s + 0.192i·27-s + 0.879·29-s − 0.362i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.9645764943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9645764943\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
| good | 5 | \( 1 + 795.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 2.62e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.00e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.10e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.96e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.35e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 1.39e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 6.21e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 3.34e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.98e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 4.49e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 4.61e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 5.76e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.06e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 7.68e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 8.82e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.01e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.28e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.72e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 3.34e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 3.64e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 2.49e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 6.99e7T + 7.83e15T^{2} \) |
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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
−10.27615227604397374208431222314, −9.002570611487808001216649375610, −8.342996556596852549684357831187, −7.46699408807431735705879864259, −6.61883467131182748952065935317, −5.44836509108504952232723331406, −4.39739754959497552024562004009, −3.19216647076179239166479872940, −2.19823897653335087359128548481, −0.831808404694199634732001724481,
0.25958960165947775047341391333, 1.25355125629210402975772836675, 3.27079568149523552546926406164, 3.74434041025626924780496797936, 4.66355204679544424586939459767, 5.89955008046624872902591769229, 7.13324831509981599747099563938, 7.973279086579427337509760267020, 8.636281728649421551908161050109, 10.01763421309462860277832541014
