| L(s) = 1 | + 10·7-s + 20·13-s + 32·19-s + 41·25-s − 2·31-s − 40·37-s − 100·43-s − 23·49-s + 152·61-s + 20·67-s + 130·73-s + 28·79-s + 200·91-s − 170·97-s + 340·103-s − 328·109-s + 17·121-s + 127-s + 131-s + 320·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 10/7·7-s + 1.53·13-s + 1.68·19-s + 1.63·25-s − 0.0645·31-s − 1.08·37-s − 2.32·43-s − 0.469·49-s + 2.49·61-s + 0.298·67-s + 1.78·73-s + 0.354·79-s + 2.19·91-s − 1.75·97-s + 3.30·103-s − 3.00·109-s + 0.140·121-s + 0.00787·127-s + 0.00763·131-s + 2.40·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.792741998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.792741998\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 41 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 254 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 914 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 782 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 238 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4382 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4889 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6062 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 76 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 1982 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13769 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 7742 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 85 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.425988106809249620743146874220, −8.723659136660937307165516926271, −8.407941981301434943798293521686, −8.367915488285622921947510688527, −7.83979932901134235429165921120, −7.41324393877948669293309893521, −6.89549980845960599620069649713, −6.55832387415150657911825506639, −6.27681701977882467786375488291, −5.32134036081736609444724001651, −5.18092298823030786504609296435, −5.16596672103773687726846325395, −4.39170551437312454250036165073, −3.80433219835183863095069479350, −3.42754001122960623869534860313, −3.02339770472018527124582886577, −2.25581915237928960410422401550, −1.53875378303095680369120804449, −1.30353875071647683524153017130, −0.63281782683444973059285064470,
0.63281782683444973059285064470, 1.30353875071647683524153017130, 1.53875378303095680369120804449, 2.25581915237928960410422401550, 3.02339770472018527124582886577, 3.42754001122960623869534860313, 3.80433219835183863095069479350, 4.39170551437312454250036165073, 5.16596672103773687726846325395, 5.18092298823030786504609296435, 5.32134036081736609444724001651, 6.27681701977882467786375488291, 6.55832387415150657911825506639, 6.89549980845960599620069649713, 7.41324393877948669293309893521, 7.83979932901134235429165921120, 8.367915488285622921947510688527, 8.407941981301434943798293521686, 8.723659136660937307165516926271, 9.425988106809249620743146874220
