L(s) = 1 | + (0.5 + 0.866i)2-s + (1 − 1.73i)3-s + (−0.499 + 0.866i)4-s + (−1.5 + 2.59i)5-s + 1.99·6-s + (2 − 3.46i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s − 3·10-s − 6·11-s + (0.999 + 1.73i)12-s + (−1 + 1.73i)13-s + 3.99·14-s + (3 + 5.19i)15-s + (−0.5 − 0.866i)16-s + (−1.5 − 2.59i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 + 0.433i)4-s + (−0.670 + 1.16i)5-s + 0.816·6-s + (0.755 − 1.30i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s − 0.948·10-s − 1.80·11-s + (0.288 + 0.499i)12-s + (−0.277 + 0.480i)13-s + 1.06·14-s + (0.774 + 1.34i)15-s + (−0.125 − 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12755 + 0.167072i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12755 + 0.167072i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-5.5 + 2.59i)T \) |
good | 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6 + 10.3i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
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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
−14.46612310158116595310787215373, −13.70109124751105375943807488161, −12.92716226973792120455014388449, −11.33939819520528070453147229241, −10.40682458920880950055875691345, −8.165443290775452655722194638782, −7.47356788247230065397830730477, −6.89372335788240693191062248397, −4.68609982432455321402057069443, −2.84776382367424505759218468873,
2.79092090144508449658333842243, 4.62481559599403455827310685414, 5.25336430560591741725632318855, 8.243035394302099928905449408459, 8.766420929132927914421575909952, 10.08308586760505866467050837192, 11.26414665020741052750068267436, 12.45115738634829876929923446628, 13.15604242257151514758203168890, 14.95653443958464231055700798311