L(s) = 1 | + 3·5-s + (0.5 + 0.866i)7-s + (1.5 + 2.59i)9-s + (−1 + 1.73i)11-s + (−2.5 − 2.59i)13-s + (3.5 + 6.06i)17-s + (1 + 1.73i)19-s + (−2 + 3.46i)23-s + 4·25-s + (−0.5 + 0.866i)29-s − 4·31-s + (1.5 + 2.59i)35-s + (−0.5 + 0.866i)37-s + (1.5 − 2.59i)41-s + (−3 − 5.19i)43-s + ⋯ |
L(s) = 1 | + 1.34·5-s + (0.188 + 0.327i)7-s + (0.5 + 0.866i)9-s + (−0.301 + 0.522i)11-s + (−0.693 − 0.720i)13-s + (0.848 + 1.47i)17-s + (0.229 + 0.397i)19-s + (−0.417 + 0.722i)23-s + 0.800·25-s + (−0.0928 + 0.160i)29-s − 0.718·31-s + (0.253 + 0.439i)35-s + (−0.0821 + 0.142i)37-s + (0.234 − 0.405i)41-s + (−0.457 − 0.792i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.138588014\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.138588014\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (2.5 + 2.59i)T \) |
good | 3 | \( 1 + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 + 7T + 53T^{2} \) |
| 59 | \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.851259231346422508549802387260, −8.965231910549266476618988421470, −7.87923823160792181457257147458, −7.43493128352852941946715329263, −6.13208981590489176415304611844, −5.52836548642965136414053969257, −4.89271489953220975788949510565, −3.56012266822761867606211845634, −2.21911335826607023183133824550, −1.65258274120730474679837401261,
0.873367360739746288101611102650, 2.14986935362268315041604883913, 3.14746800526403503142794500086, 4.41939813342562213080371176624, 5.28536065982449639448995139818, 6.10402356057140977060197784475, 6.92252919756119755386150981119, 7.61200861996304077968202548624, 8.850979557710448850730859238175, 9.585170822503347615497551168353