L(s) = 1 | + 8.04·5-s − 2·7-s − 21.7·11-s − 17.3i·13-s + 11.8i·17-s + 10.7i·19-s − 35.0i·23-s + 39.7·25-s + 11.7·29-s − 35.5·31-s − 16.0·35-s + 26i·37-s + 2.28i·41-s + 65.5i·43-s + 27.2i·47-s + ⋯ |
L(s) = 1 | + 1.60·5-s − 0.285·7-s − 1.97·11-s − 1.33i·13-s + 0.697i·17-s + 0.566i·19-s − 1.52i·23-s + 1.59·25-s + 0.405·29-s − 1.14·31-s − 0.459·35-s + 0.702i·37-s + 0.0558i·41-s + 1.52i·43-s + 0.578i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01574799694\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01574799694\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 8.04T + 25T^{2} \) |
| 7 | \( 1 + 2T + 49T^{2} \) |
| 11 | \( 1 + 21.7T + 121T^{2} \) |
| 13 | \( 1 + 17.3iT - 169T^{2} \) |
| 17 | \( 1 - 11.8iT - 289T^{2} \) |
| 19 | \( 1 - 10.7iT - 361T^{2} \) |
| 23 | \( 1 + 35.0iT - 529T^{2} \) |
| 29 | \( 1 - 11.7T + 841T^{2} \) |
| 31 | \( 1 + 35.5T + 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 2.28iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 65.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 27.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 49.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 7.52iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 65.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 84.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 84.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 75.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 48.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 106.T + 9.40e3T^{2} \) |
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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
−9.386816082897275271422454704347, −8.200198321159839156826875273790, −7.950587463124242028817275767036, −6.65130989642593595784363553735, −5.94067984853685420073133443839, −5.41922429233542176190997149373, −4.65775957482054140482880905654, −3.05207760059113523280406623091, −2.59145013733823928328275890950, −1.47289259880348247993379280572,
0.00327928874587874504303791337, 1.69911003736281196070970296017, 2.37005035229868301278338074847, 3.28181929126671357094311500046, 4.72561362205722908983732088260, 5.38027285703765102214912799146, 5.94410980922001892151398846365, 6.96480458848549069359913359184, 7.50152838447705465826267212297, 8.677185505799873097282260702885