L(s) = 1 | + (−1.5 + 0.866i)3-s + (2 − 3.46i)5-s + (1 + 1.73i)7-s + (1.5 − 2.59i)9-s + (−2.5 − 4.33i)11-s + (−1 + 1.73i)13-s + 6.92i·15-s − 3·17-s − 19-s + (−3 − 1.73i)21-s + (3 − 5.19i)23-s + (−5.49 − 9.52i)25-s + 5.19i·27-s + (−1 − 1.73i)29-s + (2 − 3.46i)31-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)3-s + (0.894 − 1.54i)5-s + (0.377 + 0.654i)7-s + (0.5 − 0.866i)9-s + (−0.753 − 1.30i)11-s + (−0.277 + 0.480i)13-s + 1.78i·15-s − 0.727·17-s − 0.229·19-s + (−0.654 − 0.377i)21-s + (0.625 − 1.08i)23-s + (−1.09 − 1.90i)25-s + 0.999i·27-s + (−0.185 − 0.321i)29-s + (0.359 − 0.622i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.838861 - 0.703887i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.838861 - 0.703887i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
good | 5 | \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.5 + 6.06i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 3T + 73T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + (-5.5 - 9.52i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.54278886195656541682018265941, −9.583293321165901040547049164017, −8.847583932990132399624278974533, −8.263135505741028390028014532381, −6.51710117776102343224362163516, −5.67748646320448688626658378536, −5.10056379857187043365488284763, −4.25765311861324647021574533751, −2.31921188868815788905969804701, −0.69051615206741144692655257586,
1.75969376071989419692407391477, 2.82042901408217682863237725691, 4.55574452533685937713230057223, 5.52696458607118756525658402236, 6.54290021446911292507568138783, 7.20725774666102267819940473836, 7.76693174108882274756112681547, 9.589289455909210751780961107214, 10.29109408642528746595551147739, 10.82904414926888945141100165622