L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 1.93·5-s + 0.999·8-s + (0.965 + 1.67i)10-s − 5.46·11-s + (1.22 + 2.12i)13-s + (−0.5 − 0.866i)16-s + (−3.15 − 5.46i)17-s + (−0.965 + 1.67i)19-s + (0.965 − 1.67i)20-s + (2.73 + 4.73i)22-s + 5.92·23-s − 1.26·25-s + (1.22 − 2.12i)26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.863·5-s + 0.353·8-s + (0.305 + 0.529i)10-s − 1.64·11-s + (0.339 + 0.588i)13-s + (−0.125 − 0.216i)16-s + (−0.765 − 1.32i)17-s + (−0.221 + 0.383i)19-s + (0.215 − 0.374i)20-s + (0.582 + 1.00i)22-s + 1.23·23-s − 0.253·25-s + (0.240 − 0.416i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7739272629\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7739272629\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.93T + 5T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.15 + 5.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.965 - 1.67i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 5.92T + 23T^{2} \) |
| 29 | \( 1 + (0.366 - 0.633i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.67 - 6.36i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.82 + 4.89i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.901 + 1.56i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.76 - 8.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.63 - 2.83i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.50 - 4.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.48 + 2.56i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.09 + 12.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + (-4.76 - 8.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.06 - 8.76i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.94 + 8.57i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.64 - 11.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.93 + 3.34i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.922373147844951496445968720300, −8.039752135510350852089350269451, −7.45803757972636129193632760132, −6.80460337369196922846627258157, −5.45612282288869084523409732175, −4.77378564313424118604515623702, −3.87609267803137715851223536615, −2.96248830534018031595033740979, −2.14064137918624919859707316315, −0.58319920131682102901245747328,
0.53614445913597701884539483424, 2.17205070070983240598044080629, 3.29862487261348060001989788854, 4.28058538657993100866328968187, 5.10430538742795413853852523029, 5.87599010802611889614189673400, 6.74902351058059415696085464015, 7.58600826995688377522626899497, 8.153074432481093104839684461014, 8.558599293221654038197322752145