L(s) = 1 | + 1.41·2-s + 1.00·4-s − 5-s + 1.41·7-s − 1.41·10-s + 11-s − 1.41·13-s + 2.00·14-s − 0.999·16-s − 1.41·17-s − 1.00·20-s + 1.41·22-s + 25-s − 2.00·26-s + 1.41·28-s − 1.41·32-s − 2.00·34-s − 1.41·35-s − 1.41·43-s + 1.00·44-s + 1.00·49-s + 1.41·50-s − 1.41·52-s − 55-s − 1.00·64-s + 1.41·65-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.00·4-s − 5-s + 1.41·7-s − 1.41·10-s + 11-s − 1.41·13-s + 2.00·14-s − 0.999·16-s − 1.41·17-s − 1.00·20-s + 1.41·22-s + 25-s − 2.00·26-s + 1.41·28-s − 1.41·32-s − 2.00·34-s − 1.41·35-s − 1.41·43-s + 1.00·44-s + 1.00·49-s + 1.41·50-s − 1.41·52-s − 55-s − 1.00·64-s + 1.41·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.569351507\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.569351507\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.41T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 1.41T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−11.63633365161304499878604028655, −10.78278788497993579819077069892, −9.252448330256898682323669948803, −8.340580974623892224952042236814, −7.30860922716506160251771233941, −6.48681841951405252696355460847, −4.96659237125259421029170422703, −4.62993254366454567203985334929, −3.65498948138900309174046932255, −2.20273546697962650221544046503,
2.20273546697962650221544046503, 3.65498948138900309174046932255, 4.62993254366454567203985334929, 4.96659237125259421029170422703, 6.48681841951405252696355460847, 7.30860922716506160251771233941, 8.340580974623892224952042236814, 9.252448330256898682323669948803, 10.78278788497993579819077069892, 11.63633365161304499878604028655