| L(s) = 1 | − 34.7i·5-s + 31.2·7-s + 57.7i·11-s − 73.2·13-s − 386. i·17-s − 115.·19-s + 548. i·23-s − 581.·25-s + 785. i·29-s − 544.·31-s − 1.08e3i·35-s + 898.·37-s + 2.58e3i·41-s − 2.00e3·43-s − 811. i·47-s + ⋯ |
| L(s) = 1 | − 1.38i·5-s + 0.636·7-s + 0.477i·11-s − 0.433·13-s − 1.33i·17-s − 0.320·19-s + 1.03i·23-s − 0.930·25-s + 0.933i·29-s − 0.566·31-s − 0.884i·35-s + 0.656·37-s + 1.54i·41-s − 1.08·43-s − 0.367i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7907594919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7907594919\) |
| \(L(3)\) |
|
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 | \( 1 + 34.7iT - 625T^{2} \) |
| 7 | \( 1 - 31.2T + 2.40e3T^{2} \) |
| 11 | \( 1 - 57.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 73.2T + 2.85e4T^{2} \) |
| 17 | \( 1 + 386. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 115.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 548. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 785. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 544.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 898.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.58e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.00e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 811. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.22e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.51e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.90e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 4.50e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.99e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.43e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.20e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 9.25e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 8.92e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 6.67e3T + 8.85e7T^{2} \) |
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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
−9.331872734049162874382277627938, −8.538220766819997779430903521165, −7.75853550831684902289399842911, −7.03650803839576328963267043591, −5.75888685210298518581840793351, −4.84057263199976663376753288467, −4.62654494427882206397879606646, −3.19741831527829844597734480194, −1.86871092172126808560526891795, −1.05628879084602569688587383570,
0.16056736074127671409665439072, 1.76137388649450411268363777048, 2.63768889757575290163255496457, 3.62793108790167369604099179008, 4.53367274649792542146443536606, 5.76979692589650749889601628400, 6.42505146981375324927960471193, 7.24944480328691352362881268471, 8.066112455277013294165499241353, 8.757176268810063791726443802238
