L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.35 + 1.07i)3-s + 1.00i·4-s + (1.25 − 1.85i)5-s + (1.72 + 0.194i)6-s + (−3.56 + 3.56i)7-s + (0.707 − 0.707i)8-s + (0.668 − 2.92i)9-s + (−2.19 + 0.421i)10-s − 3.35i·11-s + (−1.07 − 1.35i)12-s + (0.759 + 0.759i)13-s + 5.03·14-s + (0.298 + 3.86i)15-s − 1.00·16-s + (0.534 + 0.534i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.781 + 0.623i)3-s + 0.500i·4-s + (0.561 − 0.827i)5-s + (0.702 + 0.0793i)6-s + (−1.34 + 1.34i)7-s + (0.250 − 0.250i)8-s + (0.222 − 0.974i)9-s + (−0.694 + 0.133i)10-s − 1.01i·11-s + (−0.311 − 0.390i)12-s + (0.210 + 0.210i)13-s + 1.34·14-s + (0.0771 + 0.997i)15-s − 0.250·16-s + (0.129 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.555917 - 0.406058i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.555917 - 0.406058i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.35 - 1.07i)T \) |
| 5 | \( 1 + (-1.25 + 1.85i)T \) |
| 19 | \( 1 + iT \) |
good | 7 | \( 1 + (3.56 - 3.56i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.35iT - 11T^{2} \) |
| 13 | \( 1 + (-0.759 - 0.759i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.534 - 0.534i)T + 17iT^{2} \) |
| 23 | \( 1 + (-3.95 + 3.95i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.01T + 29T^{2} \) |
| 31 | \( 1 - 8.18T + 31T^{2} \) |
| 37 | \( 1 + (-5.57 + 5.57i)T - 37iT^{2} \) |
| 41 | \( 1 + 8.43iT - 41T^{2} \) |
| 43 | \( 1 + (-4.29 - 4.29i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.00 + 8.00i)T + 47iT^{2} \) |
| 53 | \( 1 + (-7.04 + 7.04i)T - 53iT^{2} \) |
| 59 | \( 1 + 4.32T + 59T^{2} \) |
| 61 | \( 1 - 1.35T + 61T^{2} \) |
| 67 | \( 1 + (-5.23 + 5.23i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.74iT - 71T^{2} \) |
| 73 | \( 1 + (5.12 + 5.12i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.256iT - 79T^{2} \) |
| 83 | \( 1 + (4.73 - 4.73i)T - 83iT^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + (9.73 - 9.73i)T - 97iT^{2} \) |
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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
−10.43722378680200132800272976478, −9.615040972317695353504966258365, −9.051755801483630821852082851597, −8.470688605381039959705648590850, −6.59011513114950582767632373732, −5.95008613303081190346055999433, −5.09964592222487442890362313547, −3.71145201177815288994613079223, −2.54767760578133885948197386434, −0.58496052096013581795242555685,
1.21571805911568411936068021391, 2.93903193352183222355418823864, 4.49598610519181413939487946264, 5.87800487912117647048127687442, 6.56684700223375882036554544886, 7.16471654292689392276478817966, 7.78958595567215396253456945996, 9.554352336170835421780583374012, 10.00053458076162407294203808252, 10.67849878147317983335677037867