L(s) = 1 | + 2i·2-s − 4·4-s + (5 − 10i)5-s − 26i·7-s − 8i·8-s + (20 + 10i)10-s + 28·11-s + 12i·13-s + 52·14-s + 16·16-s − 64i·17-s + 60·19-s + (−20 + 40i)20-s + 56i·22-s + 58i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.447 − 0.894i)5-s − 1.40i·7-s − 0.353i·8-s + (0.632 + 0.316i)10-s + 0.767·11-s + 0.256i·13-s + 0.992·14-s + 0.250·16-s − 0.913i·17-s + 0.724·19-s + (−0.223 + 0.447i)20-s + 0.542i·22-s + 0.525i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.47223 - 0.347546i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47223 - 0.347546i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-5 + 10i)T \) |
good | 7 | \( 1 + 26iT - 343T^{2} \) |
| 11 | \( 1 - 28T + 1.33e3T^{2} \) |
| 13 | \( 1 - 12iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 64iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 60T + 6.85e3T^{2} \) |
| 23 | \( 1 - 58iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 90T + 2.43e4T^{2} \) |
| 31 | \( 1 + 128T + 2.97e4T^{2} \) |
| 37 | \( 1 + 236iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 242T + 6.89e4T^{2} \) |
| 43 | \( 1 - 362iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 226iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 108iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 20T + 2.05e5T^{2} \) |
| 61 | \( 1 - 542T + 2.26e5T^{2} \) |
| 67 | \( 1 - 434iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 632iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 720T + 4.93e5T^{2} \) |
| 83 | \( 1 - 478iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 490T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3iT - 9.12e5T^{2} \) |
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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
−13.79934501469412659492758232011, −12.78133766531299122438211706657, −11.44230959184334270178854208085, −9.931513698976302874941609123232, −9.121979813686114628549284665492, −7.73717373385852751475773673662, −6.69141129736587027760913977770, −5.19060409633077467208310361828, −3.98229243459530304075045869866, −1.00308999499348308821136217698,
2.03780995823625177150951216368, 3.40197252937799934279077142547, 5.40522984126788248683214496483, 6.58374486232062297849879486054, 8.395198648954966982696387793693, 9.453491132063162949116773119033, 10.48569249280349020229035668800, 11.62579111570910799448357892943, 12.40460183519336567068946869050, 13.66177341479452577170666142652