L(s) = 1 | + (1 + i)2-s + (−3 + 3i)3-s + 2i·4-s − 6·6-s + (3 + 3i)7-s + (−2 + 2i)8-s − 9i·9-s + 12·11-s + (−6 − 6i)12-s + (12 − 12i)13-s + 6i·14-s − 4·16-s + (−12 − 12i)17-s + (9 − 9i)18-s + 20i·19-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (−1 + i)3-s + 0.5i·4-s − 6-s + (0.428 + 0.428i)7-s + (−0.250 + 0.250i)8-s − i·9-s + 1.09·11-s + (−0.5 − 0.5i)12-s + (0.923 − 0.923i)13-s + 0.428i·14-s − 0.250·16-s + (−0.705 − 0.705i)17-s + (0.5 − 0.5i)18-s + 1.05i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.702158 + 0.887215i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.702158 + 0.887215i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (3 - 3i)T - 9iT^{2} \) |
| 7 | \( 1 + (-3 - 3i)T + 49iT^{2} \) |
| 11 | \( 1 - 12T + 121T^{2} \) |
| 13 | \( 1 + (-12 + 12i)T - 169iT^{2} \) |
| 17 | \( 1 + (12 + 12i)T + 289iT^{2} \) |
| 19 | \( 1 - 20iT - 361T^{2} \) |
| 23 | \( 1 + (3 - 3i)T - 529iT^{2} \) |
| 29 | \( 1 + 30iT - 841T^{2} \) |
| 31 | \( 1 + 8T + 961T^{2} \) |
| 37 | \( 1 + (-48 - 48i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 48T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-27 + 27i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (27 + 27i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-12 + 12i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 60iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 32T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-3 - 3i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 48T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-12 + 12i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 40iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (93 - 93i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 30iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (12 + 12i)T + 9.40e3iT^{2} \) |
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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
−15.65081437613277756070886873702, −14.85675972353831863899237310532, −13.49346587648384673340222799839, −11.94484439296439243756063310825, −11.21824167426596634666356975027, −9.822762371023839079051358572040, −8.322712636911506303332342674312, −6.35504837536380960253135998046, −5.30158693937804148156623983341, −3.95109862782897319960374742534,
1.42504353568295107066098519823, 4.29252318897959727144555749919, 6.08538527085902077423606947587, 7.01340998185642322714023690803, 8.997678127128207322577605709863, 10.98086827193740151289353130432, 11.46712333871302629580932963337, 12.63517194630713297048588129221, 13.57085292982982797615314154746, 14.66072378348971495484848169715