L(s) = 1 | + (−1.29 + 1.15i)3-s + (0.186 − 0.186i)5-s − 7-s + (0.350 − 2.97i)9-s + (−4.29 − 4.29i)11-s + (2.73 − 2.73i)13-s + (−0.0267 + 0.455i)15-s + 4.98i·17-s + (3.09 + 3.09i)19-s + (1.29 − 1.15i)21-s + 3.55i·23-s + 4.93i·25-s + (2.97 + 4.26i)27-s + (3.75 + 3.75i)29-s + 6.58i·31-s + ⋯ |
L(s) = 1 | + (−0.747 + 0.664i)3-s + (0.0833 − 0.0833i)5-s − 0.377·7-s + (0.116 − 0.993i)9-s + (−1.29 − 1.29i)11-s + (0.759 − 0.759i)13-s + (−0.00691 + 0.117i)15-s + 1.20i·17-s + (0.710 + 0.710i)19-s + (0.282 − 0.251i)21-s + 0.742i·23-s + 0.986i·25-s + (0.572 + 0.819i)27-s + (0.697 + 0.697i)29-s + 1.18i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.461 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7186411813\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7186411813\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.29 - 1.15i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (-0.186 + 0.186i)T - 5iT^{2} \) |
| 11 | \( 1 + (4.29 + 4.29i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.73 + 2.73i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.98iT - 17T^{2} \) |
| 19 | \( 1 + (-3.09 - 3.09i)T + 19iT^{2} \) |
| 23 | \( 1 - 3.55iT - 23T^{2} \) |
| 29 | \( 1 + (-3.75 - 3.75i)T + 29iT^{2} \) |
| 31 | \( 1 - 6.58iT - 31T^{2} \) |
| 37 | \( 1 + (4.82 + 4.82i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + (-5.79 + 5.79i)T - 43iT^{2} \) |
| 47 | \( 1 + 3.22T + 47T^{2} \) |
| 53 | \( 1 + (7.95 - 7.95i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.18 - 3.18i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.17 - 3.17i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.39 + 2.39i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.06iT - 71T^{2} \) |
| 73 | \( 1 + 2.32iT - 73T^{2} \) |
| 79 | \( 1 - 6.89iT - 79T^{2} \) |
| 83 | \( 1 + (-1.12 + 1.12i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.26044357601939046551776193548, −9.019722232164932883575252050321, −8.430245729506213772176343578059, −7.45509682243198153119311411176, −6.27706270016015643210262839224, −5.59743057310609090799194245927, −5.15606800019648184265554713654, −3.57334693380451746760576090168, −3.26708682344010419360605261699, −1.21438068366085880523198592734,
0.35653746195376890190581684677, 1.97759587486153750887752544796, 2.87506086839896757684988648466, 4.59167687509130083367957207888, 5.02203876116084642592064161912, 6.24107617691760578267336311867, 6.80805722488859487028566081838, 7.58188361188166545804477803134, 8.347118108308398035782977372281, 9.582134308735634342820857189673