L(s) = 1 | + (1.05 + 1.82i)3-s + (1.46 − 2.54i)5-s + (2.64 + 0.102i)7-s + (−0.711 + 1.23i)9-s + (2.90 + 5.02i)11-s − 1.53·13-s + 6.17·15-s + (−2.84 − 4.92i)17-s + (−1.65 + 2.86i)19-s + (2.59 + 4.92i)21-s + (1.45 − 2.51i)23-s + (−1.80 − 3.13i)25-s + 3.31·27-s − 1.53·29-s + (3.33 + 5.76i)31-s + ⋯ |
L(s) = 1 | + (0.607 + 1.05i)3-s + (0.656 − 1.13i)5-s + (0.999 + 0.0387i)7-s + (−0.237 + 0.410i)9-s + (0.875 + 1.51i)11-s − 0.425·13-s + 1.59·15-s + (−0.690 − 1.19i)17-s + (−0.379 + 0.656i)19-s + (0.565 + 1.07i)21-s + (0.303 − 0.525i)23-s + (−0.361 − 0.626i)25-s + 0.638·27-s − 0.285·29-s + (0.598 + 1.03i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.587661190\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.587661190\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.64 - 0.102i)T \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + (-1.05 - 1.82i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.46 + 2.54i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.90 - 5.02i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 + (2.84 + 4.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.65 - 2.86i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.45 + 2.51i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.53T + 29T^{2} \) |
| 31 | \( 1 + (-3.33 - 5.76i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.67 + 9.82i)T + (-18.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 + (4.48 - 7.77i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.43 + 2.49i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.87 + 8.45i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.28 - 5.68i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.17 + 7.23i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + (1.01 + 1.75i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.15 - 8.93i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.68T + 83T^{2} \) |
| 89 | \( 1 + (-2.40 + 4.16i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4.61T + 97T^{2} \) |
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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
−9.505056391563856315016470791911, −9.306031008012579772834478659254, −8.553231949600302978625192373229, −7.57099691123751233371773993481, −6.53202728868329633238089069734, −5.10258950308942680569202890327, −4.70813742718271093813622727470, −4.07077212644061623437511264621, −2.46758787604752786415621510646, −1.43716854723840830600126745371,
1.33798107659015906016964354010, 2.28244193828532667806018655921, 3.15269246528844365409759310223, 4.43312971633502187290956534818, 5.86395746512718004887272684376, 6.49018902849467586631104093437, 7.17638596484593453397490840518, 8.202369412272465672330346543534, 8.570799829501255677800017870716, 9.683501032713337250286858188993