L(s) = 1 | + 1.29i·2-s + i·3-s + 0.335·4-s + i·5-s − 1.29·6-s + (2.23 − 1.42i)7-s + 3.01i·8-s − 9-s − 1.29·10-s + (0.810 − 3.21i)11-s + 0.335i·12-s + 3.24·13-s + (1.83 + 2.87i)14-s − 15-s − 3.21·16-s + 7.53·17-s + ⋯ |
L(s) = 1 | + 0.912i·2-s + 0.577i·3-s + 0.167·4-s + 0.447i·5-s − 0.526·6-s + (0.843 − 0.537i)7-s + 1.06i·8-s − 0.333·9-s − 0.408·10-s + (0.244 − 0.969i)11-s + 0.0967i·12-s + 0.898·13-s + (0.490 + 0.769i)14-s − 0.258·15-s − 0.804·16-s + 1.82·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.235472858\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.235472858\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-2.23 + 1.42i)T \) |
| 11 | \( 1 + (-0.810 + 3.21i)T \) |
good | 2 | \( 1 - 1.29iT - 2T^{2} \) |
| 13 | \( 1 - 3.24T + 13T^{2} \) |
| 17 | \( 1 - 7.53T + 17T^{2} \) |
| 19 | \( 1 - 2.10T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 - 7.87iT - 29T^{2} \) |
| 31 | \( 1 + 2.42iT - 31T^{2} \) |
| 37 | \( 1 + 8.46T + 37T^{2} \) |
| 41 | \( 1 + 6.82T + 41T^{2} \) |
| 43 | \( 1 + 8.43iT - 43T^{2} \) |
| 47 | \( 1 + 2.05iT - 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 9.43iT - 59T^{2} \) |
| 61 | \( 1 - 8.80T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 3.80T + 73T^{2} \) |
| 79 | \( 1 + 5.78iT - 79T^{2} \) |
| 83 | \( 1 + 9.24T + 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 2.85iT - 97T^{2} \) |
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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
−10.40128808700195997387270516135, −8.860444328046643595930379468667, −8.356487551592060219234834378189, −7.49332052989806110175394209505, −6.81609363970966028296960035485, −5.60978850932530759387601559640, −5.39485498538382974287152949500, −3.88940421777983141514212500241, −3.11749326390254808701323573961, −1.43224319920750064967865218399,
1.21062056686265827038325850761, 1.83681857451892784101618706026, 3.04090871455892364777175360847, 4.10268313523919485658927313728, 5.27520661549567623944776426085, 6.11429979978655954008914156754, 7.20472010446106840821482878277, 7.929743497086705796630851112130, 8.748811346404783366435777542718, 9.783664451601508786093499412042