L(s) = 1 | + (−1.29 + 0.569i)2-s + (−0.908 − 1.47i)3-s + (1.35 − 1.47i)4-s + (2.01 + 1.39i)6-s + 2.50i·7-s + (−0.908 + 2.67i)8-s + (−1.35 + 2.67i)9-s + 3.36·11-s + (−3.40 − 0.652i)12-s + 3.70·13-s + (−1.42 − 3.24i)14-s + (−0.350 − 3.98i)16-s − 7.63i·17-s + (0.222 − 4.23i)18-s + 0.440i·19-s + ⋯ |
L(s) = 1 | + (−0.915 + 0.402i)2-s + (−0.524 − 0.851i)3-s + (0.675 − 0.737i)4-s + (0.822 + 0.568i)6-s + 0.948i·7-s + (−0.321 + 0.947i)8-s + (−0.450 + 0.892i)9-s + 1.01·11-s + (−0.982 − 0.188i)12-s + 1.02·13-s + (−0.382 − 0.868i)14-s + (−0.0876 − 0.996i)16-s − 1.85i·17-s + (0.0523 − 0.998i)18-s + 0.100i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.778945 - 0.0740756i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.778945 - 0.0740756i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.29 - 0.569i)T \) |
| 3 | \( 1 + (0.908 + 1.47i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2.50iT - 7T^{2} \) |
| 11 | \( 1 - 3.36T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 + 7.63iT - 17T^{2} \) |
| 19 | \( 1 - 0.440iT - 19T^{2} \) |
| 23 | \( 1 - 5.17T + 23T^{2} \) |
| 29 | \( 1 - 2.27iT - 29T^{2} \) |
| 31 | \( 1 + 3.39iT - 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 + 3.07iT - 41T^{2} \) |
| 43 | \( 1 - 8.40iT - 43T^{2} \) |
| 47 | \( 1 + 3.63T + 47T^{2} \) |
| 53 | \( 1 - 2.27iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 + 5.45iT - 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 1.29T + 73T^{2} \) |
| 79 | \( 1 - 5.01iT - 79T^{2} \) |
| 83 | \( 1 + 1.81T + 83T^{2} \) |
| 89 | \( 1 - 5.35iT - 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−11.47115321824282791654295439432, −11.07862329985997940412151614617, −9.506236095744510470494011335355, −8.888829912734996298598335045379, −7.84218910616527782838753101613, −6.82806264343977017480908339940, −6.09841396852147598464354501930, −5.09611165470014568834411359405, −2.66495544682658220965434713518, −1.12929481694610625423474453754,
1.19060063888702337925324625850, 3.50947399904895256247682526073, 4.20442727113141712215161408166, 6.06940820365282048234876388241, 6.86671433091559353436995876064, 8.283895679272102895574413471345, 9.070307344759043302380416121572, 10.05333236210151170342241123640, 10.80456376765805439916506631329, 11.30371986833736184063425546242