| L(s) = 1 | + 2-s − 4-s + 1.41i·5-s − 2.82i·7-s − 3·8-s + 1.41i·10-s + 4.24i·13-s − 2.82i·14-s − 16-s − 6·17-s − 1.41i·20-s + 8.48i·23-s + 2.99·25-s + 4.24i·26-s + 2.82i·28-s − 8·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.5·4-s + 0.632i·5-s − 1.06i·7-s − 1.06·8-s + 0.447i·10-s + 1.17i·13-s − 0.755i·14-s − 0.250·16-s − 1.45·17-s − 0.316i·20-s + 1.76i·23-s + 0.599·25-s + 0.832i·26-s + 0.534i·28-s − 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5655334362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5655334362\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 8.48iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 1.41iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 7.07iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + 7.07iT - 73T^{2} \) |
| 79 | \( 1 + 8.48iT - 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−10.28386302674145506190150135339, −9.270767444191115457050762504746, −8.821530432246395056577364210723, −7.33527517311975019436416413494, −6.97378168104924839469547757038, −5.89082107297916174387086381749, −4.89830335170751826937406022994, −3.97494312672525015030931134646, −3.44197402475290713933458150761, −1.87541829871856476581948085132,
0.19026246820860583551012464840, 2.20554667488558793374068410950, 3.30344458171355219951648282808, 4.41770025287563331225164272071, 5.20396691560226240479717611412, 5.76871584237633525626608958480, 6.78313108307094394755798447436, 8.121550675525604134772370416068, 8.937983384080349493656413036513, 9.087382126779537228013400247292
