L(s) = 1 | + (1.94 − 1.11i)5-s + (−0.576 + 0.576i)7-s − 0.279i·11-s + (−1.37 − 1.37i)13-s + (−1.92 − 1.92i)17-s − 4.53i·19-s + (−0.707 + 0.707i)23-s + (2.53 − 4.31i)25-s − 9.30·29-s − 8.69·31-s + (−0.478 + 1.76i)35-s + (−2.34 + 2.34i)37-s + 5.29i·41-s + (5.71 + 5.71i)43-s + (−6.03 − 6.03i)47-s + ⋯ |
L(s) = 1 | + (0.867 − 0.496i)5-s + (−0.218 + 0.218i)7-s − 0.0843i·11-s + (−0.380 − 0.380i)13-s + (−0.465 − 0.465i)17-s − 1.04i·19-s + (−0.147 + 0.147i)23-s + (0.506 − 0.862i)25-s − 1.72·29-s − 1.56·31-s + (−0.0809 + 0.297i)35-s + (−0.384 + 0.384i)37-s + 0.826i·41-s + (0.871 + 0.871i)43-s + (−0.879 − 0.879i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5297956384\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5297956384\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.94 + 1.11i)T \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (0.576 - 0.576i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.279iT - 11T^{2} \) |
| 13 | \( 1 + (1.37 + 1.37i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.92 + 1.92i)T + 17iT^{2} \) |
| 19 | \( 1 + 4.53iT - 19T^{2} \) |
| 29 | \( 1 + 9.30T + 29T^{2} \) |
| 31 | \( 1 + 8.69T + 31T^{2} \) |
| 37 | \( 1 + (2.34 - 2.34i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.29iT - 41T^{2} \) |
| 43 | \( 1 + (-5.71 - 5.71i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.03 + 6.03i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.60 - 7.60i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.99T + 59T^{2} \) |
| 61 | \( 1 + 6.61T + 61T^{2} \) |
| 67 | \( 1 + (-7.67 + 7.67i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.41iT - 71T^{2} \) |
| 73 | \( 1 + (3.26 + 3.26i)T + 73iT^{2} \) |
| 79 | \( 1 - 13.5iT - 79T^{2} \) |
| 83 | \( 1 + (-2.76 + 2.76i)T - 83iT^{2} \) |
| 89 | \( 1 - 5.27T + 89T^{2} \) |
| 97 | \( 1 + (8.87 - 8.87i)T - 97iT^{2} \) |
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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
−8.076642777948927347308987532985, −7.32917702316389391364204598810, −6.54310948127868766210694105138, −5.77257569257068495786032704016, −5.15553015187203693291099465513, −4.44510396426401498587836353157, −3.28056580107650354207404736307, −2.43008447041372928358709912415, −1.52752180807015214411553776752, −0.13344208009578896517534724791,
1.69162037128608008376476195135, 2.22361819335874980904203844143, 3.47653834658132866973731515709, 4.04130937042502114866333724218, 5.30558406400573342309824350226, 5.74781713948759859795525099523, 6.63847526620402781539341542002, 7.18715460819660752368887615196, 7.942168167151651641596901737499, 8.966259135753573188022146201084