L(s) = 1 | − 2.23i·5-s − 10.0i·7-s + 9.12i·11-s − 5.62·13-s − 14.7i·17-s − 9.77i·19-s + (15.4 + 17.0i)23-s − 5.00·25-s + 22.5·29-s + 9.14·31-s − 22.4·35-s + 0.327i·37-s + 64.1·41-s − 42.8i·43-s + 11.5·47-s + ⋯ |
L(s) = 1 | − 0.447i·5-s − 1.43i·7-s + 0.829i·11-s − 0.432·13-s − 0.866i·17-s − 0.514i·19-s + (0.671 + 0.741i)23-s − 0.200·25-s + 0.776·29-s + 0.294·31-s − 0.641·35-s + 0.00884i·37-s + 1.56·41-s − 0.995i·43-s + 0.245·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.622285746\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.622285746\) |
\(L(2)\) |
|
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 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-15.4 - 17.0i)T \) |
good | 7 | \( 1 + 10.0iT - 49T^{2} \) |
| 11 | \( 1 - 9.12iT - 121T^{2} \) |
| 13 | \( 1 + 5.62T + 169T^{2} \) |
| 17 | \( 1 + 14.7iT - 289T^{2} \) |
| 19 | \( 1 + 9.77iT - 361T^{2} \) |
| 29 | \( 1 - 22.5T + 841T^{2} \) |
| 31 | \( 1 - 9.14T + 961T^{2} \) |
| 37 | \( 1 - 0.327iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 64.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 42.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 11.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 63.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 51.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 46.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 59.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 11.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 67.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 84.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 40.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 4.77iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 41.8iT - 9.40e3T^{2} \) |
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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
−7.76446346194526081811575080513, −7.21365125635266323987295723464, −6.82201728694156261990635260514, −5.63569071497752556179897376307, −4.75263460540826972997852469502, −4.36735727247172358156369428542, −3.38603972427096966961918950396, −2.36045896460810269776344501073, −1.18637045228753581945523964246, −0.37752428888860868917258057869,
1.13594383296206972689743326824, 2.44127541038520048556141588898, 2.84231866131160278048562062420, 3.91575970141672501541366333613, 4.86539148484287838140389899308, 5.80900197527475170742391691887, 6.11485700970745669049964646114, 6.98556240549761799025761837134, 8.002282736278531968473746715757, 8.479108571589054513041194919596