L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.943 + 1.29i)5-s + (−0.951 + 0.309i)7-s + (0.309 − 0.951i)8-s − 1.60i·10-s + (1.01 + 3.15i)11-s + (−1.22 + 1.68i)13-s + (0.951 + 0.309i)14-s + (−0.809 + 0.587i)16-s + (−4.11 + 2.98i)17-s + (−6.93 − 2.25i)19-s + (−0.943 + 1.29i)20-s + (1.03 − 3.15i)22-s − 0.571i·23-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (0.422 + 0.580i)5-s + (−0.359 + 0.116i)7-s + (0.109 − 0.336i)8-s − 0.507i·10-s + (0.306 + 0.951i)11-s + (−0.338 + 0.466i)13-s + (0.254 + 0.0825i)14-s + (−0.202 + 0.146i)16-s + (−0.997 + 0.724i)17-s + (−1.59 − 0.516i)19-s + (−0.211 + 0.290i)20-s + (0.220 − 0.671i)22-s − 0.119i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3279220532\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3279220532\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.951 - 0.309i)T \) |
| 11 | \( 1 + (-1.01 - 3.15i)T \) |
good | 5 | \( 1 + (-0.943 - 1.29i)T + (-1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (1.22 - 1.68i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (4.11 - 2.98i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (6.93 + 2.25i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 0.571iT - 23T^{2} \) |
| 29 | \( 1 + (-0.282 - 0.867i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (5.53 + 4.02i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.0690 + 0.212i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.429 - 1.32i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 7.30iT - 43T^{2} \) |
| 47 | \( 1 + (-2.53 - 0.824i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.49 - 2.05i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.53 - 0.823i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (4.63 + 6.37i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 + (1.60 + 2.20i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.42 - 2.73i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.22 - 1.68i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (2.76 - 2.00i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 1.76iT - 89T^{2} \) |
| 97 | \( 1 + (-5.45 - 3.96i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.966778380479916116446542031949, −9.143005396911413081794840982475, −8.611569139685919903585268035337, −7.43201482896196211324433044333, −6.70307064467642149028704433845, −6.14977906282576845299426006092, −4.64962000254181698670465165245, −3.88268249385314233064611806234, −2.47373804055963844400235819096, −1.93602270898845560952604393493,
0.15085834624703189589964654865, 1.58467436334207655502906440344, 2.88168006665629606290724652373, 4.18714492408738892729984016044, 5.21915492811562641202713439715, 6.03014974099974366611014788155, 6.74041962638164648550279504196, 7.66912852283779741230253371406, 8.684999676646633195759755071087, 8.984393706526426486263887945934