L(s) = 1 | − i·2-s − 4-s − 0.618i·7-s + i·8-s − 9-s − 1.61·11-s − 1.61i·13-s − 0.618·14-s + 16-s + i·18-s − 0.618·19-s + 1.61i·22-s − 1.61i·23-s − 1.61·26-s + 0.618i·28-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s − 0.618i·7-s + i·8-s − 9-s − 1.61·11-s − 1.61i·13-s − 0.618·14-s + 16-s + i·18-s − 0.618·19-s + 1.61i·22-s − 1.61i·23-s − 1.61·26-s + 0.618i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5354403360\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5354403360\) |
\(L(1)\) |
|
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 | \( 1 \) |
good | 3 | \( 1 + T^{2} \) |
| 7 | \( 1 + 0.618iT - T^{2} \) |
| 11 | \( 1 + 1.61T + T^{2} \) |
| 13 | \( 1 + 1.61iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 0.618T + T^{2} \) |
| 23 | \( 1 + 1.61iT - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.618iT - T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.61iT - T^{2} \) |
| 53 | \( 1 - 0.618iT - T^{2} \) |
| 59 | \( 1 + 0.618T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - 1.61T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−10.13721911445573437365205234392, −8.972715820975204788441834558145, −8.146253669486179286464895710017, −7.67948355963812845910812034145, −6.03309163557421023811755445472, −5.27803604498195023920795062366, −4.35950311828659390244930121243, −3.04839952210451431850416556123, −2.50502165784129229413176958115, −0.48110794230330819991389489200,
2.24649263784152407964293391754, 3.55732012453145747468403924025, 4.82706710304148869006178771927, 5.51017068280432819385862507952, 6.27597403158564102147500296468, 7.27168070509826751643936382108, 8.092314340119241540856755082354, 8.810160839691377597651914720580, 9.446498523307904755142083271714, 10.43203885754029139844855445320