L(s) = 1 | + (0.5 + 0.866i)5-s + (−2 + 1.73i)7-s + (1.5 − 2.59i)11-s + (−0.5 + 0.866i)13-s + (1.5 + 2.59i)17-s + (2.5 − 4.33i)19-s + (−0.5 − 0.866i)23-s + (2 − 3.46i)25-s + (4.5 + 7.79i)29-s − 4·31-s + (−2.5 − 0.866i)35-s + (−2.5 + 4.33i)37-s + (3.5 − 6.06i)41-s + (1.5 + 2.59i)43-s + 8·47-s + ⋯ |
L(s) = 1 | + (0.223 + 0.387i)5-s + (−0.755 + 0.654i)7-s + (0.452 − 0.783i)11-s + (−0.138 + 0.240i)13-s + (0.363 + 0.630i)17-s + (0.573 − 0.993i)19-s + (−0.104 − 0.180i)23-s + (0.400 − 0.692i)25-s + (0.835 + 1.44i)29-s − 0.718·31-s + (−0.422 − 0.146i)35-s + (−0.410 + 0.711i)37-s + (0.546 − 0.946i)41-s + (0.228 + 0.396i)43-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.708529099\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.708529099\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.5 + 6.06i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 - 2.59i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (-6.5 - 11.2i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.985913729479718967831380398549, −8.233174962578292023469683553017, −7.11633652885132385851304770990, −6.59503516280162287222229234414, −5.85777717362417224918877599729, −5.14614998307676859804294781221, −3.98919237570411684373141638999, −3.11240433503660135053255715929, −2.44675504335956544698178453999, −1.01104755304490151672534901495,
0.65696023828015177518718461867, 1.80732380135353407547154501953, 3.03455879373448703715894101996, 3.87324837390155450402425278720, 4.67411412605140816136168538629, 5.61272125178578237686975088797, 6.29528527917720156835495088659, 7.33180214280573488193473806053, 7.57119411049803624747703675143, 8.732933934379693464505594422958