L(s) = 1 | + (1.55 + 2.56i)3-s + 1.34i·7-s + (−4.17 + 7.97i)9-s − 3.66i·11-s − 7.34i·13-s + 6.31·17-s + 30.7·19-s + (−3.44 + 2.08i)21-s + 26.2·23-s + (−26.9 + 1.68i)27-s + 40.9i·29-s + 9.97·31-s + (9.40 − 5.69i)33-s + 39.4i·37-s + (18.8 − 11.4i)39-s + ⋯ |
L(s) = 1 | + (0.517 + 0.855i)3-s + 0.191i·7-s + (−0.463 + 0.886i)9-s − 0.333i·11-s − 0.564i·13-s + 0.371·17-s + 1.61·19-s + (−0.164 + 0.0993i)21-s + 1.14·23-s + (−0.998 + 0.0624i)27-s + 1.41i·29-s + 0.321·31-s + (0.285 − 0.172i)33-s + 1.06i·37-s + (0.483 − 0.292i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0806 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0806 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.279437379\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.279437379\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.55 - 2.56i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.34iT - 49T^{2} \) |
| 11 | \( 1 + 3.66iT - 121T^{2} \) |
| 13 | \( 1 + 7.34iT - 169T^{2} \) |
| 17 | \( 1 - 6.31T + 289T^{2} \) |
| 19 | \( 1 - 30.7T + 361T^{2} \) |
| 23 | \( 1 - 26.2T + 529T^{2} \) |
| 29 | \( 1 - 40.9iT - 841T^{2} \) |
| 31 | \( 1 - 9.97T + 961T^{2} \) |
| 37 | \( 1 - 39.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 68.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 79.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 38.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 39.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 64.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 53.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 136. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 74.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 79.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 0.0506T + 6.88e3T^{2} \) |
| 89 | \( 1 - 22.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 35.0iT - 9.40e3T^{2} \) |
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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
−9.658416326864753502648714217384, −9.108292114000202233255871005221, −8.182306699379505581911640089419, −7.56584352022837064239920910527, −6.36740974736341708768535916650, −5.24440939854702655183571555627, −4.78765531370897497825568890198, −3.26615545182295544918216910881, −3.03111324061963720957316589225, −1.27055820101116393647253637701,
0.69654892830893169624705314828, 1.85971577623307949472938704929, 2.96051223891873799296734418761, 3.91272881126656351292824721252, 5.16223332853322889708077100223, 6.11254112201448733470975424859, 7.14263037920483933689731585815, 7.49730188341628593804664609428, 8.489165393628687658839318097542, 9.308311101957422348661710431676