L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s − 10.3i·7-s − 7.99·8-s − 10.3i·11-s − 18·13-s + (−18 − 10.3i)14-s + (−8 + 13.8i)16-s + 10·17-s + 13.8i·19-s + (−18 − 10.3i)22-s + 6.92i·23-s + (−18 + 31.1i)26-s + (−36 + 20.7i)28-s + 36·29-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 1.48i·7-s − 0.999·8-s − 0.944i·11-s − 1.38·13-s + (−1.28 − 0.742i)14-s + (−0.5 + 0.866i)16-s + 0.588·17-s + 0.729i·19-s + (−0.818 − 0.472i)22-s + 0.301i·23-s + (−0.692 + 1.19i)26-s + (−1.28 + 0.742i)28-s + 1.24·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.005180289\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.005180289\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 10.3iT - 49T^{2} \) |
| 11 | \( 1 + 10.3iT - 121T^{2} \) |
| 13 | \( 1 + 18T + 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 19 | \( 1 - 13.8iT - 361T^{2} \) |
| 23 | \( 1 - 6.92iT - 529T^{2} \) |
| 29 | \( 1 - 36T + 841T^{2} \) |
| 31 | \( 1 + 6.92iT - 961T^{2} \) |
| 37 | \( 1 + 54T + 1.36e3T^{2} \) |
| 41 | \( 1 + 18T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 26T + 2.80e3T^{2} \) |
| 59 | \( 1 - 31.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 74T + 3.72e3T^{2} \) |
| 67 | \( 1 + 41.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 36T + 5.32e3T^{2} \) |
| 79 | \( 1 - 90.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 90.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 18T + 7.92e3T^{2} \) |
| 97 | \( 1 - 72T + 9.40e3T^{2} \) |
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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
−9.801281212016367616404375153480, −8.628667826421034459072840981055, −7.63145445143976453830458523809, −6.71899796186069682518214152659, −5.60758832718049720137127617911, −4.69765888267163942743976470368, −3.78005164997067597649200975304, −2.93957805555843669080057086622, −1.44138269675064367895320038094, −0.27263540512954913450864968701,
2.25712449346531607530462782551, 3.14069843552697171708698947382, 4.75263145445140604838458051010, 5.06793166461503951398854127958, 6.16778990892014727661048587078, 6.99910128620688543507556032620, 7.81409981894127600642898685582, 8.740590340222497032252455340999, 9.387976789205114897970052659484, 10.23089801128313510943441740399