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
−10.67849878147317983335677037867, −10.00053458076162407294203808252, −9.554352336170835421780583374012, −7.78958595567215396253456945996, −7.16471654292689392276478817966, −6.56684700223375882036554544886, −5.87800487912117647048127687442, −4.49598610519181413939487946264, −2.93903193352183222355418823864, −1.21571805911568411936068021391,
0.58496052096013581795242555685, 2.54767760578133885948197386434, 3.71145201177815288994613079223, 5.09964592222487442890362313547, 5.95008613303081190346055999433, 6.59011513114950582767632373732, 8.470688605381039959705648590850, 9.051755801483630821852082851597, 9.615040972317695353504966258365, 10.43722378680200132800272976478