L(s) = 1 | + (−1.07 + 2.60i)3-s + (−0.707 + 0.292i)5-s + (−1.68 + 1.68i)7-s + (−3.50 − 3.50i)9-s + (−0.334 − 0.808i)11-s + (−1.09 − 0.451i)13-s − 2.15i·15-s + 0.224i·17-s + (−2.87 − 1.19i)19-s + (−2.57 − 6.21i)21-s + (3.68 + 3.68i)23-s + (−3.12 + 3.12i)25-s + (5.09 − 2.11i)27-s + (−2.34 + 5.66i)29-s − 6.82·31-s + ⋯ |
L(s) = 1 | + (−0.623 + 1.50i)3-s + (−0.316 + 0.130i)5-s + (−0.637 + 0.637i)7-s + (−1.16 − 1.16i)9-s + (−0.100 − 0.243i)11-s + (−0.302 − 0.125i)13-s − 0.557i·15-s + 0.0545i·17-s + (−0.660 − 0.273i)19-s + (−0.561 − 1.35i)21-s + (0.768 + 0.768i)23-s + (−0.624 + 0.624i)25-s + (0.981 − 0.406i)27-s + (−0.435 + 1.05i)29-s − 1.22·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0143528 + 0.581269i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0143528 + 0.581269i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (1.07 - 2.60i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (0.707 - 0.292i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (1.68 - 1.68i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.334 + 0.808i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (1.09 + 0.451i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 0.224iT - 17T^{2} \) |
| 19 | \( 1 + (2.87 + 1.19i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-3.68 - 3.68i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.34 - 5.66i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 + (-9.87 + 4.09i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-6.37 - 6.37i)T + 41iT^{2} \) |
| 43 | \( 1 + (-1.90 - 4.60i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 0.542iT - 47T^{2} \) |
| 53 | \( 1 + (-3.91 - 9.46i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (3.36 - 1.39i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (0.398 - 0.962i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-1.48 + 3.57i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-5.39 + 5.39i)T - 71iT^{2} \) |
| 73 | \( 1 + (5.15 + 5.15i)T + 73iT^{2} \) |
| 79 | \( 1 + 8.39iT - 79T^{2} \) |
| 83 | \( 1 + (-11.2 - 4.64i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (5.92 - 5.92i)T - 89iT^{2} \) |
| 97 | \( 1 + 4.19T + 97T^{2} \) |
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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
−12.31305223742707720301256247430, −11.15023893977731832033054917188, −10.81331102695242290845016027204, −9.421377848994240575817596609760, −9.218909927439222102446370210289, −7.59645096895636054627195447815, −6.11722547191182112625398136519, −5.30716561134383202819928653108, −4.13754776064998006319109655548, −3.03686114827668590796831728618,
0.47603890690648962035794115000, 2.26865181478770828047109475372, 4.14103730336549341602736423694, 5.70825175237864965079937692741, 6.68343224903928130117092901922, 7.38005552482027553449139512083, 8.291885830500063297673195085368, 9.704366979032638783018967856992, 10.85630621181541105777427077508, 11.72187701945647862391043115520