L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s − 4·14-s + 16-s − 2·17-s − 19-s + 20-s − 8·23-s + 25-s − 4·28-s − 2·29-s + 8·31-s + 32-s − 2·34-s − 4·35-s − 10·37-s − 38-s + 40-s + 6·41-s − 8·43-s − 8·46-s + 9·49-s + 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s − 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.229·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s − 0.755·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.676·35-s − 1.64·37-s − 0.162·38-s + 0.158·40-s + 0.937·41-s − 1.21·43-s − 1.17·46-s + 9/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9234642246\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9234642246\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.74595779943509, −12.36267754127394, −11.90392057477914, −11.56065188969808, −10.82202357834640, −10.33380343260002, −10.01689792717395, −9.719796042215182, −8.915754918628894, −8.776431691955762, −7.926615609382348, −7.559855842993456, −6.824003690754307, −6.423783240862273, −6.274159986382906, −5.645074876654449, −5.188735921676416, −4.479750645619336, −4.040707592284930, −3.516635060391095, −2.990164929114425, −2.502864969080117, −1.924084143623934, −1.270889774996424, −0.2147212251213839,
0.2147212251213839, 1.270889774996424, 1.924084143623934, 2.502864969080117, 2.990164929114425, 3.516635060391095, 4.040707592284930, 4.479750645619336, 5.188735921676416, 5.645074876654449, 6.274159986382906, 6.423783240862273, 6.824003690754307, 7.559855842993456, 7.926615609382348, 8.776431691955762, 8.915754918628894, 9.719796042215182, 10.01689792717395, 10.33380343260002, 10.82202357834640, 11.56065188969808, 11.90392057477914, 12.36267754127394, 12.74595779943509