L(s) = 1 | − 0.662i·3-s + (−1.31 − 1.81i)5-s + (−1.19 − 2.35i)7-s + 2.56·9-s − 3.09i·11-s + 4.66·13-s + (−1.19 + 0.868i)15-s + 2.04·17-s + 5.60·19-s + (−1.56 + 0.794i)21-s − 1.87·23-s + (−1.56 + 4.74i)25-s − 3.68i·27-s + 3.56·29-s + 8.74·31-s + ⋯ |
L(s) = 1 | − 0.382i·3-s + (−0.586 − 0.810i)5-s + (−0.453 − 0.891i)7-s + 0.853·9-s − 0.932i·11-s + 1.29·13-s + (−0.309 + 0.224i)15-s + 0.496·17-s + 1.28·19-s + (−0.340 + 0.173i)21-s − 0.390·23-s + (−0.312 + 0.949i)25-s − 0.708i·27-s + 0.661·29-s + 1.57·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.766787807\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.766787807\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.31 + 1.81i)T \) |
| 7 | \( 1 + (1.19 + 2.35i)T \) |
good | 3 | \( 1 + 0.662iT - 3T^{2} \) |
| 11 | \( 1 + 3.09iT - 11T^{2} \) |
| 13 | \( 1 - 4.66T + 13T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 - 3.56T + 29T^{2} \) |
| 31 | \( 1 - 8.74T + 31T^{2} \) |
| 37 | \( 1 + 3.70iT - 37T^{2} \) |
| 41 | \( 1 + 8.48iT - 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 + 0.290iT - 47T^{2} \) |
| 53 | \( 1 - 9.49iT - 53T^{2} \) |
| 59 | \( 1 + 8.05T + 59T^{2} \) |
| 61 | \( 1 - 6.45iT - 61T^{2} \) |
| 67 | \( 1 + 2.39T + 67T^{2} \) |
| 71 | \( 1 + 9.65iT - 71T^{2} \) |
| 73 | \( 1 - 4.09T + 73T^{2} \) |
| 79 | \( 1 + 1.35iT - 79T^{2} \) |
| 83 | \( 1 + 12.4iT - 83T^{2} \) |
| 89 | \( 1 - 2.82iT - 89T^{2} \) |
| 97 | \( 1 + 6.14T + 97T^{2} \) |
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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
−8.684659738735241780519179090250, −7.920288175208010003357369440871, −7.40611898594696390278646177893, −6.46748510726175475659390968820, −5.73167792406968171788026732417, −4.61950377764722901070102333756, −3.85002005175472822388528901612, −3.17295642080240315632708895542, −1.31967179039167339369526006998, −0.75510969171832719649193077467,
1.39006613202520299843275480685, 2.78391383616666584851625406599, 3.48456400249924601473675126700, 4.37600660567123908079326143686, 5.26632202621746814365559819529, 6.39722243190360389790239448025, 6.78047938823168710424834738041, 7.85885474082652392512194486154, 8.366995822360693377439389157689, 9.627625281309863246590695848450