| L(s) = 1 | − 5.79i·5-s + 8.39·7-s + 14.6i·11-s − 21.1·13-s − 7.76i·17-s − 24.3·19-s − 14.6i·23-s − 8.58·25-s − 35.4i·29-s − 8·31-s − 48.6i·35-s − 60.5·37-s + 33.6i·41-s − 9.17·43-s − 16.9i·47-s + ⋯ |
| L(s) = 1 | − 1.15i·5-s + 1.19·7-s + 1.33i·11-s − 1.63·13-s − 0.456i·17-s − 1.28·19-s − 0.638i·23-s − 0.343·25-s − 1.22i·29-s − 0.258·31-s − 1.38i·35-s − 1.63·37-s + 0.820i·41-s − 0.213·43-s − 0.361i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4730211925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4730211925\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.79iT - 25T^{2} \) |
| 7 | \( 1 - 8.39T + 49T^{2} \) |
| 11 | \( 1 - 14.6iT - 121T^{2} \) |
| 13 | \( 1 + 21.1T + 169T^{2} \) |
| 17 | \( 1 + 7.76iT - 289T^{2} \) |
| 19 | \( 1 + 24.3T + 361T^{2} \) |
| 23 | \( 1 + 14.6iT - 529T^{2} \) |
| 29 | \( 1 + 35.4iT - 841T^{2} \) |
| 31 | \( 1 + 8T + 961T^{2} \) |
| 37 | \( 1 + 60.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 9.17T + 1.84e3T^{2} \) |
| 47 | \( 1 + 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 25.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 61.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 13T + 3.72e3T^{2} \) |
| 67 | \( 1 + 21.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 101. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 98.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 103. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 134. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 75.1T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.010779605900445820841720744160, −8.252685070396755841499054982167, −7.55786673811306509432844834571, −6.73275375283748688485826261340, −5.32797316019796860284801868814, −4.70993661251794357499616899028, −4.33643729064058213127158256660, −2.40296578133448690859050432650, −1.65776504755189490951745443898, −0.12482688691228236603032515157,
1.71600817307626589376200493582, 2.73828161527383416946395665745, 3.70529335975245434359924577407, 4.87964426280892170657407950193, 5.65924812262009628101147141622, 6.73318344679522735896331241232, 7.35890908295454562780808953460, 8.251745546783180776470016114235, 8.900121166193303260136698590494, 10.09076258241388486561138342919
