L(s) = 1 | − 1.73i·2-s + 1.73i·3-s − 0.999·4-s + 2.99·6-s + (−2 − 1.73i)7-s − 1.73i·8-s − 2.99·9-s − 3.46i·11-s − 1.73i·12-s + (−2.99 + 3.46i)14-s − 5·16-s + 6·17-s + 5.19i·18-s − 3.46i·19-s + (2.99 − 3.46i)21-s − 5.99·22-s + ⋯ |
L(s) = 1 | − 1.22i·2-s + 0.999i·3-s − 0.499·4-s + 1.22·6-s + (−0.755 − 0.654i)7-s − 0.612i·8-s − 0.999·9-s − 1.04i·11-s − 0.499i·12-s + (−0.801 + 0.925i)14-s − 1.25·16-s + 1.45·17-s + 1.22i·18-s − 0.794i·19-s + (0.654 − 0.755i)21-s − 1.27·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.484869 - 1.06133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.484869 - 1.06133i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−10.47212270757445484131359350354, −9.977409902473546394359670122745, −9.225366481388965813566345209463, −8.162909626428848534701027055385, −6.76658042870903684333855826712, −5.71045363267218455269869724935, −4.36762313245250054001214211565, −3.46471743218789052078383852691, −2.79266163078902945878075225686, −0.68009710406838669263553681590,
1.88052529093017967831170943536, 3.25680015186414588701100659352, 5.17036163867387101142343237463, 5.87330229878917318829235879409, 6.69271271891388967550066955704, 7.47384913185477231515421335849, 8.111675385615524935608401307889, 9.133659500153687179359894363497, 10.04587377959947312920249244919, 11.45237090828949478474383705406