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
−11.72187701945647862391043115520, −10.85630621181541105777427077508, −9.704366979032638783018967856992, −8.291885830500063297673195085368, −7.38005552482027553449139512083, −6.68343224903928130117092901922, −5.70825175237864965079937692741, −4.14103730336549341602736423694, −2.26865181478770828047109475372, −0.47603890690648962035794115000,
3.03686114827668590796831728618, 4.13754776064998006319109655548, 5.30716561134383202819928653108, 6.11722547191182112625398136519, 7.59645096895636054627195447815, 9.218909927439222102446370210289, 9.421377848994240575817596609760, 10.81331102695242290845016027204, 11.15023893977731832033054917188, 12.31305223742707720301256247430