| L(s) = 1 | − 27i·3-s − 183. i·5-s + 284.·7-s − 729·9-s − 5.59e3i·11-s + 1.42e4i·13-s − 4.96e3·15-s + 6.68e3·17-s − 2.55e4i·19-s − 7.68e3i·21-s + 3.82e4·23-s + 4.42e4·25-s + 1.96e4i·27-s − 1.59e5i·29-s − 3.15e5·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.658i·5-s + 0.313·7-s − 0.333·9-s − 1.26i·11-s + 1.79i·13-s − 0.380·15-s + 0.330·17-s − 0.854i·19-s − 0.181i·21-s + 0.655·23-s + 0.566·25-s + 0.192i·27-s − 1.21i·29-s − 1.89·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.274037927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.274037927\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27iT \) |
| good | 5 | \( 1 + 183. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 284.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.59e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.42e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 6.68e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.55e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 3.82e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.59e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 3.15e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.45e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 2.98e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.12e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 4.44e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.77e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.80e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 5.13e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 3.66e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 3.21e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.10e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.20e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.29e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 7.41e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.06e6T + 8.07e13T^{2} \) |
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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
−9.231468252304263476431756497446, −8.960650978981292548364666026407, −7.84221079116676479364028979765, −6.88591598407219091741866111593, −5.90189199898051045926330758455, −4.87606442495508253324204878791, −3.75503465944506021370131347395, −2.34285674502435119920369528813, −1.25735834770133757220055626655, −0.27069807906997984333906510603,
1.34117118440812256165548156001, 2.74966464416628680064779857585, 3.60680165365498513189284815900, 4.90714944550632380329610195504, 5.65242747360100448267520985207, 7.00938909576290785981613610760, 7.76027563625285938220663221489, 8.822997775788361696741995623495, 9.989414904790244099623497790997, 10.44973054373025134175673857243
