L(s) = 1 | + i·2-s + 1.20i·3-s − 4-s + (1.77 + 1.36i)5-s − 1.20·6-s + 4.33i·7-s − i·8-s + 1.55·9-s + (−1.36 + 1.77i)10-s + 0.276·11-s − 1.20i·12-s − 4.16i·13-s − 4.33·14-s + (−1.63 + 2.12i)15-s + 16-s − 2.65i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.692i·3-s − 0.5·4-s + (0.792 + 0.609i)5-s − 0.489·6-s + 1.63i·7-s − 0.353i·8-s + 0.519·9-s + (−0.431 + 0.560i)10-s + 0.0834·11-s − 0.346i·12-s − 1.15i·13-s − 1.15·14-s + (−0.422 + 0.549i)15-s + 0.250·16-s − 0.644i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.487136 + 1.43235i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.487136 + 1.43235i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-1.77 - 1.36i)T \) |
| 43 | \( 1 + iT \) |
good | 3 | \( 1 - 1.20iT - 3T^{2} \) |
| 7 | \( 1 - 4.33iT - 7T^{2} \) |
| 11 | \( 1 - 0.276T + 11T^{2} \) |
| 13 | \( 1 + 4.16iT - 13T^{2} \) |
| 17 | \( 1 + 2.65iT - 17T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 23 | \( 1 - 1.82iT - 23T^{2} \) |
| 29 | \( 1 + 8.13T + 29T^{2} \) |
| 31 | \( 1 - 0.868T + 31T^{2} \) |
| 37 | \( 1 + 5.72iT - 37T^{2} \) |
| 41 | \( 1 - 4.91T + 41T^{2} \) |
| 47 | \( 1 + 7.94iT - 47T^{2} \) |
| 53 | \( 1 + 1.31iT - 53T^{2} \) |
| 59 | \( 1 - 4.68T + 59T^{2} \) |
| 61 | \( 1 - 5.55T + 61T^{2} \) |
| 67 | \( 1 + 7.06iT - 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 4.44iT - 73T^{2} \) |
| 79 | \( 1 + 3.37T + 79T^{2} \) |
| 83 | \( 1 - 9.61iT - 83T^{2} \) |
| 89 | \( 1 - 8.27T + 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−11.36656316538805131157576193962, −10.38993610123724291912813325242, −9.510965388959224269589739624350, −9.048647454327489548402657237015, −7.82825129154186761377041679154, −6.73871932919303431279315854434, −5.59416710365025301451699780060, −5.26109418355224458079036626883, −3.60263096606264280688480810279, −2.30198201544671123352662708969,
1.07592533165659049076516236618, 1.99486796780707926286156248128, 3.93075861511654830080920282205, 4.63158447307650221561587946019, 6.20145653083436478877982539120, 7.07175192787216514154352989073, 8.054505612090371179269248080834, 9.226051945619262478794314780499, 9.992354396168651780008480658891, 10.73725417968086438108137927972