L(s) = 1 | + 2-s + 4-s + (1.67 − 1.47i)5-s + 8-s + (1.67 − 1.47i)10-s + 6.33i·11-s − 1.05·13-s + 16-s + 4.63i·17-s − 6.17i·19-s + (1.67 − 1.47i)20-s + 6.33i·22-s + 7.04·23-s + (0.624 − 4.96i)25-s − 1.05·26-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.749 − 0.661i)5-s + 0.353·8-s + (0.530 − 0.467i)10-s + 1.90i·11-s − 0.292·13-s + 0.250·16-s + 1.12i·17-s − 1.41i·19-s + (0.374 − 0.330i)20-s + 1.34i·22-s + 1.46·23-s + (0.124 − 0.992i)25-s − 0.206·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.630908184\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.630908184\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.67 + 1.47i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6.33iT - 11T^{2} \) |
| 13 | \( 1 + 1.05T + 13T^{2} \) |
| 17 | \( 1 - 4.63iT - 17T^{2} \) |
| 19 | \( 1 + 6.17iT - 19T^{2} \) |
| 23 | \( 1 - 7.04T + 23T^{2} \) |
| 29 | \( 1 - 2.98iT - 29T^{2} \) |
| 31 | \( 1 - 6.31iT - 31T^{2} \) |
| 37 | \( 1 - 3.47iT - 37T^{2} \) |
| 41 | \( 1 - 6.97T + 41T^{2} \) |
| 43 | \( 1 + 2.58iT - 43T^{2} \) |
| 47 | \( 1 + 8.15iT - 47T^{2} \) |
| 53 | \( 1 + 8.87T + 53T^{2} \) |
| 59 | \( 1 - 0.904T + 59T^{2} \) |
| 61 | \( 1 - 10.1iT - 61T^{2} \) |
| 67 | \( 1 - 9.83iT - 67T^{2} \) |
| 71 | \( 1 - 14.4iT - 71T^{2} \) |
| 73 | \( 1 - 8.36T + 73T^{2} \) |
| 79 | \( 1 + 5.46T + 79T^{2} \) |
| 83 | \( 1 + 14.6iT - 83T^{2} \) |
| 89 | \( 1 - 3.83T + 89T^{2} \) |
| 97 | \( 1 - 5.87T + 97T^{2} \) |
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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
−8.586936908423392843976775935188, −7.32683006959335946280249730108, −6.99016897116485157231895257405, −6.18462482260486097327858954312, −5.13087346434462221741974015520, −4.87251983006374691052136620764, −4.12223180198298767040918483327, −2.88838159263375348296442108715, −2.09141886947848629942814852663, −1.23725403482395490318923360247,
0.833537898368165664992503946175, 2.13969981173652123602138683645, 3.04650085169734337011107301227, 3.46684256698959377124047017877, 4.65057652252037170175594890409, 5.51600062393798692383607532210, 6.03828976787811697350393674132, 6.55382972461870095002508170897, 7.57045753764247791208077713795, 8.060189194084126811376390387486