L(s) = 1 | + (1.60 − 2.77i)3-s + (−1.45 − 2.52i)5-s + (−0.195 − 0.338i)7-s + (−3.63 − 6.29i)9-s + (4.62 − 2.67i)11-s + (−2.74 + 4.76i)13-s − 9.32·15-s − 4.03i·17-s + (−1.77 + 1.02i)19-s − 1.25·21-s + (3.40 − 1.96i)23-s + (−1.73 + 3.00i)25-s − 13.6·27-s + (4.52 − 2.61i)29-s + (−5.25 + 3.03i)31-s + ⋯ |
L(s) = 1 | + (0.924 − 1.60i)3-s + (−0.650 − 1.12i)5-s + (−0.0737 − 0.127i)7-s + (−1.21 − 2.09i)9-s + (1.39 − 0.805i)11-s + (−0.762 + 1.32i)13-s − 2.40·15-s − 0.979i·17-s + (−0.408 + 0.235i)19-s − 0.272·21-s + (0.710 − 0.410i)23-s + (−0.346 + 0.600i)25-s − 2.62·27-s + (0.840 − 0.485i)29-s + (−0.943 + 0.544i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.815356381\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.815356381\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 79 | \( 1 + (-2.66 - 8.47i)T \) |
good | 3 | \( 1 + (-1.60 + 2.77i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.45 + 2.52i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.195 + 0.338i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.62 + 2.67i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.74 - 4.76i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.03iT - 17T^{2} \) |
| 19 | \( 1 + (1.77 - 1.02i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.40 + 1.96i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.52 + 2.61i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.25 - 3.03i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.16 - 4.71i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.64iT - 41T^{2} \) |
| 43 | \( 1 + (5.59 - 9.68i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.568 - 0.983i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.76 - 3.90i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.41 + 5.91i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 10.6iT - 61T^{2} \) |
| 67 | \( 1 + 7.02iT - 67T^{2} \) |
| 71 | \( 1 - 2.42T + 71T^{2} \) |
| 73 | \( 1 + (-4.34 - 7.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 83 | \( 1 + (5.07 - 2.92i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 5.86T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−9.147720159739923219153287699346, −8.353914743439764548022475182365, −7.81963539517376711296727429137, −6.69322360686051722085661306150, −6.51916461362594091530464170128, −4.88499419952610135781209815580, −3.93469037672408440957039483458, −2.83677590960876549592711009156, −1.58724610167957304273106644451, −0.70375229622632369161714081223,
2.33553889992509824794231862437, 3.27061898135490998951262627841, 3.86836514370943272265286856170, 4.66997655060880883886299962535, 5.78170962365859302144697202018, 7.04067719920967489549298390202, 7.71682340816406410545187933075, 8.680957961896488785011554009431, 9.326403200308886000509696847189, 10.15068155020408708444010241338