L(s) = 1 | + (−1 − 1.41i)3-s + (1 − 2i)5-s + (−0.414 + 0.414i)7-s + (−1.00 + 2.82i)9-s − 4.82i·11-s + (−1.82 − 1.82i)13-s + (−3.82 + 0.585i)15-s + (−3.82 − 3.82i)17-s + 4.82i·19-s + (1 + 0.171i)21-s + (1.58 − 1.58i)23-s + (−3 − 4i)25-s + (5.00 − 1.41i)27-s + 7.65·29-s + 5.65·31-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.816i)3-s + (0.447 − 0.894i)5-s + (−0.156 + 0.156i)7-s + (−0.333 + 0.942i)9-s − 1.45i·11-s + (−0.507 − 0.507i)13-s + (−0.988 + 0.151i)15-s + (−0.928 − 0.928i)17-s + 1.10i·19-s + (0.218 + 0.0374i)21-s + (0.330 − 0.330i)23-s + (−0.600 − 0.800i)25-s + (0.962 − 0.272i)27-s + 1.42·29-s + 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.528680 - 0.799024i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.528680 - 0.799024i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 5 | \( 1 + (-1 + 2i)T \) |
good | 7 | \( 1 + (0.414 - 0.414i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.82iT - 11T^{2} \) |
| 13 | \( 1 + (1.82 + 1.82i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.82 + 3.82i)T + 17iT^{2} \) |
| 19 | \( 1 - 4.82iT - 19T^{2} \) |
| 23 | \( 1 + (-1.58 + 1.58i)T - 23iT^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + (0.171 - 0.171i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (2.41 + 2.41i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.41 - 6.41i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3 + 3i)T - 53iT^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + (-4.07 + 4.07i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.48iT - 71T^{2} \) |
| 73 | \( 1 + (-6.65 - 6.65i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.82iT - 79T^{2} \) |
| 83 | \( 1 + (5.24 - 5.24i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.34T + 89T^{2} \) |
| 97 | \( 1 + (-1 + i)T - 97iT^{2} \) |
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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
−12.00003287636609261388784649831, −11.04696859900534597233339511088, −9.955005065899607055799268894553, −8.688698113463902045504923939393, −8.007464440794589597600333635888, −6.56522091801796313853129072653, −5.74440843936315810179120356496, −4.75237944573905987417453971246, −2.67274229363317024227521916753, −0.849697606789767347319049905355,
2.44633282563865214618409515269, 4.07976335334290891779966422438, 5.06975492303816037231165127328, 6.52130409254316779265758767795, 7.04321969015559544984800931797, 8.829071249201966031866485214267, 9.867818722441321499550516902177, 10.36185647206653693729868290610, 11.34127138126336127988795485988, 12.22874148580091925975493762269