L(s) = 1 | + (−1.87 + 3.25i)5-s + 4.75·7-s + 4.36·11-s + (−0.5 − 0.866i)13-s + (3.06 − 5.30i)17-s + (0.694 + 4.30i)19-s + (−1.87 − 3.25i)23-s + (−4.56 − 7.90i)25-s + (2.69 + 4.66i)29-s + 7.36·31-s + (−8.94 + 15.4i)35-s − 7.12·37-s + (−1.75 + 3.04i)41-s + (0.379 − 0.657i)43-s + (−3 − 5.19i)47-s + ⋯ |
L(s) = 1 | + (−0.840 + 1.45i)5-s + 1.79·7-s + 1.31·11-s + (−0.138 − 0.240i)13-s + (0.743 − 1.28i)17-s + (0.159 + 0.987i)19-s + (−0.391 − 0.678i)23-s + (−0.912 − 1.58i)25-s + (0.500 + 0.866i)29-s + 1.32·31-s + (−1.51 + 2.61i)35-s − 1.17·37-s + (−0.274 + 0.475i)41-s + (0.0578 − 0.100i)43-s + (−0.437 − 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.945080170\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.945080170\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-0.694 - 4.30i)T \) |
good | 5 | \( 1 + (1.87 - 3.25i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 4.75T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.06 + 5.30i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.87 + 3.25i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.69 - 4.66i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.36T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + (1.75 - 3.04i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.379 + 0.657i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.57 - 4.45i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.63 + 8.03i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.25 - 7.37i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.684 + 1.18i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.82 - 10.0i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.25 - 3.91i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.07 - 12.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.90T + 83T^{2} \) |
| 89 | \( 1 + (-4.18 - 7.24i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.82 + 10.0i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.972160998576949139109787076720, −8.602999749841791683827203685452, −8.074954826826714272107238239848, −7.23538270313467435043580921827, −6.71694453175250487800194156024, −5.49369817148241165793102213369, −4.50124359781183489364474120124, −3.66318181835504861490790906913, −2.64228846187652217405153783509, −1.28898180626818330580976499530,
1.02771656043443994500905779604, 1.76666267177002707174778107922, 3.73283721723274827514032231770, 4.47288433553394648561383533638, 4.98355793472549819679874330248, 6.02985381817192567098938745587, 7.30479455571430664432478451628, 8.097257066277226682124131724611, 8.530259303339463906471112908466, 9.178724253221013078920270372312