L(s) = 1 | − 899. i·5-s + 907.·7-s + 2.19e4i·11-s + 2.08e4·13-s + 5.49e4i·17-s − 1.12e5·19-s − 3.64e5i·23-s − 4.18e5·25-s − 4.93e5i·29-s − 9.47e5·31-s − 8.16e5i·35-s − 2.31e6·37-s + 3.36e5i·41-s − 1.64e5·43-s − 1.21e6i·47-s + ⋯ |
L(s) = 1 | − 1.43i·5-s + 0.377·7-s + 1.49i·11-s + 0.730·13-s + 0.657i·17-s − 0.861·19-s − 1.30i·23-s − 1.07·25-s − 0.697i·29-s − 1.02·31-s − 0.543i·35-s − 1.23·37-s + 0.118i·41-s − 0.0480·43-s − 0.249i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.4133446162\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4133446162\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 907.T \) |
good | 5 | \( 1 + 899. iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 2.19e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.08e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.49e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.12e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 3.64e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.93e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 9.47e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.31e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.36e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.64e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 1.21e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 2.50e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.27e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.01e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 7.53e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 4.21e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.47e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 1.95e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 2.01e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.15e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 9.91e7T + 7.83e15T^{2} \) |
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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
−10.89308675502760857984914834005, −9.935597379088728372658784411528, −8.825261574478323199231087400196, −8.310307412319677933655806212906, −7.06903138775570110460038190414, −5.79932141718572417681909093396, −4.68948071877759906999074675168, −4.08106782300973884549707072318, −2.11645703437457627308618045026, −1.25597391849202029730914831216,
0.086539206666428555650084594174, 1.63869680012242266053083543562, 3.02307442500337916306354011275, 3.68918042958340598088620173073, 5.39323018405936891593283595256, 6.33290608028587996346745492202, 7.23803722288364701991785634768, 8.297185802010610227474198359098, 9.291040308304766026820165641142, 10.74827532772450260759052783720