| L(s) = 1 | + 724. i·2-s − 5.24e5·4-s + 4.76e6i·5-s + 9.59e6·7-s − 3.79e8i·8-s − 3.45e9·10-s − 5.16e9i·11-s + 2.71e11·13-s + 6.94e9i·14-s + 2.74e11·16-s + 1.75e12i·17-s − 1.06e13·19-s − 2.50e12i·20-s + 3.73e12·22-s − 2.41e13i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.488i·5-s + 0.0339·7-s − 0.353i·8-s − 0.345·10-s − 0.199i·11-s + 1.96·13-s + 0.0240i·14-s + 0.250·16-s + 0.868i·17-s − 1.74·19-s − 0.244i·20-s + 0.140·22-s − 0.582i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(1.772486932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.772486932\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 724. iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.76e6iT - 9.53e13T^{2} \) |
| 7 | \( 1 - 9.59e6T + 7.97e16T^{2} \) |
| 11 | \( 1 + 5.16e9iT - 6.72e20T^{2} \) |
| 13 | \( 1 - 2.71e11T + 1.90e22T^{2} \) |
| 17 | \( 1 - 1.75e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 + 1.06e13T + 3.75e25T^{2} \) |
| 23 | \( 1 + 2.41e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 + 7.49e14iT - 1.76e29T^{2} \) |
| 31 | \( 1 + 5.73e13T + 6.71e29T^{2} \) |
| 37 | \( 1 + 4.21e15T + 2.31e31T^{2} \) |
| 41 | \( 1 - 8.67e15iT - 1.80e32T^{2} \) |
| 43 | \( 1 - 5.90e14T + 4.67e32T^{2} \) |
| 47 | \( 1 + 2.92e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 - 9.03e16iT - 3.05e34T^{2} \) |
| 59 | \( 1 + 6.95e17iT - 2.61e35T^{2} \) |
| 61 | \( 1 - 3.85e17T + 5.08e35T^{2} \) |
| 67 | \( 1 + 1.83e18T + 3.32e36T^{2} \) |
| 71 | \( 1 + 4.69e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 5.37e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + 7.56e18T + 8.96e37T^{2} \) |
| 83 | \( 1 + 2.47e19iT - 2.40e38T^{2} \) |
| 89 | \( 1 - 1.20e18iT - 9.72e38T^{2} \) |
| 97 | \( 1 - 2.06e18T + 5.43e39T^{2} \) |
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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
−11.16143162519972406452094617615, −10.35885060902889350610891262159, −8.763843495745569129317053870414, −8.122771756818135709315749558741, −6.51347607522431624392135866703, −6.07448459248118621824112192209, −4.41444977020892022029483046898, −3.42556687340564955631512934761, −1.83653577090962605477141872304, −0.43200757441492148657794431588,
0.898650115662765804979160260451, 1.76211350222392969978825696039, 3.19794200158842326449160004846, 4.24537401639403190199095028219, 5.42161756300907841529704951613, 6.77084784347029078244218943732, 8.449536288734534665159672438284, 9.035021017516786841976609705167, 10.51516908917982656964263927636, 11.25960372458390856475711832274
