L(s) = 1 | + (−2 + 2i)3-s − 5i·9-s + 6·11-s + (4 + 4i)17-s + 2i·19-s + (4 + 4i)27-s + (−12 + 12i)33-s − 6·41-s + (−6 + 6i)43-s + 7i·49-s − 16·51-s + (−4 − 4i)57-s + 6i·59-s + (6 + 6i)67-s + (12 − 12i)73-s + ⋯ |
L(s) = 1 | + (−1.15 + 1.15i)3-s − 1.66i·9-s + 1.80·11-s + (0.970 + 0.970i)17-s + 0.458i·19-s + (0.769 + 0.769i)27-s + (−2.08 + 2.08i)33-s − 0.937·41-s + (−0.914 + 0.914i)43-s + i·49-s − 2.24·51-s + (−0.529 − 0.529i)57-s + 0.781i·59-s + (0.733 + 0.733i)67-s + (1.40 − 1.40i)73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.661154 + 0.835404i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.661154 + 0.835404i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (2 - 2i)T - 3iT^{2} \) |
| 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (-4 - 4i)T + 17iT^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + (6 - 6i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + (-6 - 6i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-12 + 12i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (2 - 2i)T - 83iT^{2} \) |
| 89 | \( 1 - 18iT - 89T^{2} \) |
| 97 | \( 1 + (-12 - 12i)T + 97iT^{2} \) |
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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
−10.48722080064354338540539401990, −9.791874234471254702659467099495, −9.138740062090640023502522260446, −8.054130074138071708319466951205, −6.65291589380958311300919344950, −6.07799681313080001687520879052, −5.16783668080927167766382205173, −4.14300174892925885753815429607, −3.53134546627626128718760516226, −1.31437944960221481870748300333,
0.73105651618821827495685605173, 1.82287663019988428222672353195, 3.51183439168114773091687953246, 4.89005584491219131574238981638, 5.73905525244797566231259606060, 6.74451280268357617460245481820, 7.01009412911764965826068222557, 8.150771758520677918582215325569, 9.201203727483320988190480055985, 10.09268022359926029866307532480