L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 2·12-s − 4·13-s + 16-s + 6·17-s − 18-s − 2·19-s + 2·24-s − 5·25-s + 4·26-s + 4·27-s + 6·29-s − 32-s − 6·34-s + 36-s − 2·37-s + 2·38-s + 8·39-s − 6·41-s − 8·43-s + 12·47-s − 2·48-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s + 0.408·24-s − 25-s + 0.784·26-s + 0.769·27-s + 1.11·29-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s + 1.28·39-s − 0.937·41-s − 1.21·43-s + 1.75·47-s − 0.288·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 31 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.05395240570469, −13.68515981057049, −12.65458521663913, −12.52559608500756, −11.97022994784845, −11.53488553874039, −11.29286023113862, −10.31411922095635, −10.14706031756953, −9.980915574172858, −9.062998653021335, −8.601635685390647, −7.986382643206533, −7.534127972729851, −6.940964683028485, −6.505655561942159, −5.876602265350605, −5.429338049456188, −4.974335334086180, −4.317463547268125, −3.510372057991844, −2.867569043658598, −2.183213515687798, −1.409140320022969, −0.6773102373911092, 0,
0.6773102373911092, 1.409140320022969, 2.183213515687798, 2.867569043658598, 3.510372057991844, 4.317463547268125, 4.974335334086180, 5.429338049456188, 5.876602265350605, 6.505655561942159, 6.940964683028485, 7.534127972729851, 7.986382643206533, 8.601635685390647, 9.062998653021335, 9.980915574172858, 10.14706031756953, 10.31411922095635, 11.29286023113862, 11.53488553874039, 11.97022994784845, 12.52559608500756, 12.65458521663913, 13.68515981057049, 14.05395240570469