L(s) = 1 | + (1 + 2i)5-s + 3·9-s + 4i·13-s − 8i·17-s + (−3 + 4i)25-s − 10·29-s − 12i·37-s − 10·41-s + (3 + 6i)45-s + 7·49-s − 4i·53-s + 10·61-s + (−8 + 4i)65-s + 16i·73-s + 9·81-s + ⋯ |
L(s) = 1 | + (0.447 + 0.894i)5-s + 9-s + 1.10i·13-s − 1.94i·17-s + (−0.600 + 0.800i)25-s − 1.85·29-s − 1.97i·37-s − 1.56·41-s + (0.447 + 0.894i)45-s + 49-s − 0.549i·53-s + 1.28·61-s + (−0.992 + 0.496i)65-s + 1.87i·73-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20672 + 0.284869i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20672 + 0.284869i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 12iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−13.13387627056618102443435959454, −11.82256781444945886246706336516, −11.00391439577334505958100210466, −9.836389702397532262256903701307, −9.191948760713857735786646205932, −7.34659225820132395138253365593, −6.86219003417993493021307753474, −5.35231253026681953104433630583, −3.85170305742133163906454997877, −2.16476520096918511412058198152,
1.61473517356191733655870362226, 3.81810438310669041777406398309, 5.14698111002753304188953804645, 6.25934352275442499995645379753, 7.75263392663315316171910690580, 8.671209308200170326934608018083, 9.893153518764802002543209859693, 10.56903725538474578921268476792, 12.07543356892081038834846161312, 12.99135949620404659688672522647