| L(s) = 1 | − 4.11·5-s + 1.10·11-s − 3.60i·13-s + 11.9·25-s + 7.82i·41-s − 12.4i·43-s − 10.0i·47-s + 7·49-s − 4.53·55-s − 15.3·59-s + 7.21i·61-s + 14.8i·65-s + 16.1i·71-s + 10.3·79-s − 13.1·83-s + ⋯ |
| L(s) = 1 | − 1.84·5-s + 0.332·11-s − 0.999i·13-s + 2.38·25-s + 1.22i·41-s − 1.90i·43-s − 1.46i·47-s + 49-s − 0.611·55-s − 1.99·59-s + 0.923i·61-s + 1.84i·65-s + 1.91i·71-s + 1.16·79-s − 1.44·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1218658700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1218658700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + 3.60iT \) |
| good | 5 | \( 1 + 4.11T + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 1.10T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 7.82iT - 41T^{2} \) |
| 43 | \( 1 + 12.4iT - 43T^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 15.3T + 59T^{2} \) |
| 61 | \( 1 - 7.21iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 16.1iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 4.31iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.505522527318589431545346644967, −8.171927445508278769523258796262, −7.34074568139984916586751794658, −6.90415107114947592389614771773, −5.77579348773274750576478508554, −4.93351070999681900676847567325, −4.07479397007334556476489828190, −3.52897150222595612340868031496, −2.65007669068733817453234899537, −1.03649552554361146134259607881,
0.04494986556306540218941186219, 1.38229470808509158539189587927, 2.79869823013697582934599615640, 3.66885071683757103010375159462, 4.31430714193572141204334352809, 4.88195233352655162117108766697, 6.16771781754638651972984765615, 6.84170109904022141995370196765, 7.63298970951322365321062609779, 8.006853577248549746011465088285
