L(s) = 1 | − 1.41i·2-s + (2.22 + 2.01i)3-s − 2.00·4-s + (2.84 − 3.14i)6-s − 2.64·7-s + 2.82i·8-s + (0.882 + 8.95i)9-s − 6.10i·11-s + (−4.44 − 4.02i)12-s − 6.09·13-s + 3.74i·14-s + 4.00·16-s − 7.30i·17-s + (12.6 − 1.24i)18-s − 31.7·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.740 + 0.671i)3-s − 0.500·4-s + (0.474 − 0.523i)6-s − 0.377·7-s + 0.353i·8-s + (0.0980 + 0.995i)9-s − 0.554i·11-s + (−0.370 − 0.335i)12-s − 0.468·13-s + 0.267i·14-s + 0.250·16-s − 0.429i·17-s + (0.703 − 0.0693i)18-s − 1.67·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.170375636\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.170375636\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (-2.22 - 2.01i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 11 | \( 1 + 6.10iT - 121T^{2} \) |
| 13 | \( 1 + 6.09T + 169T^{2} \) |
| 17 | \( 1 + 7.30iT - 289T^{2} \) |
| 19 | \( 1 + 31.7T + 361T^{2} \) |
| 23 | \( 1 + 33.8iT - 529T^{2} \) |
| 29 | \( 1 + 41.0iT - 841T^{2} \) |
| 31 | \( 1 - 42.9T + 961T^{2} \) |
| 37 | \( 1 - 18.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 36.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 17.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 45.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 4.22iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 16.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 17.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 113. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 105.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 74.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 76.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 26.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 120.T + 9.40e3T^{2} \) |
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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
−9.604090925362759532442860024681, −8.540255973513976492269653913748, −8.299642864339613888524733839156, −6.93858788730892154466610196184, −5.86309121561453744626436191857, −4.61436672306690624966176197706, −4.06929736456785505571523339981, −2.86655447722820634649907807331, −2.23089384020955371477626641105, −0.31966498125183382222029477193,
1.43983653315003959806612625042, 2.70832161265529994882003613141, 3.83030057527291008071822603591, 4.83139063846257573192493916626, 6.13093260473104386794490368588, 6.69738707577250122636482248798, 7.55669050566280375919230527059, 8.209998332456938740963528659309, 9.066064946822361028686328226767, 9.707647635779623977005343578692