L(s) = 1 | − 1.41·2-s + 2.00·4-s − 2.23i·5-s + (−1.41 − 2.23i)7-s − 2.82·8-s − 3·9-s + 3.16i·10-s − 2·11-s + 4.47i·13-s + (2.00 + 3.16i)14-s + 4.00·16-s + 4.24·18-s − 6.32i·19-s − 4.47i·20-s + 2.82·22-s − 8.48·23-s + ⋯ |
L(s) = 1 | − 1.00·2-s + 1.00·4-s − 0.999i·5-s + (−0.534 − 0.845i)7-s − 1.00·8-s − 9-s + 1.00i·10-s − 0.603·11-s + 1.24i·13-s + (0.534 + 0.845i)14-s + 1.00·16-s + 1.00·18-s − 1.45i·19-s − 1.00i·20-s + 0.603·22-s − 1.76·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.108135 - 0.373280i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.108135 - 0.373280i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (1.41 + 2.23i)T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 6.32iT - 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 12.6iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 13.4iT - 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 - 6.32iT - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 12.6iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−11.42835192833543389256186529465, −10.38074833927312987992390787229, −9.418656418145160266469632624221, −8.735865030706980147132889349152, −7.77574494747414507068019994680, −6.71661080634473726664643227949, −5.58633281163088331963535669494, −4.05048519308636927800139171088, −2.31572285969836195690290922794, −0.36932729754291788699717431392,
2.44880743892934855220194940368, 3.24947152112251740281402331041, 5.91663821221873493431851192087, 6.08598336037165351299571389172, 7.84501915921938188901581207127, 8.149453468914424772475614206002, 9.620168084647959289414087546848, 10.19541933510476648234642130152, 11.15487576584123192635637705302, 11.95451140847756025672039007863