| L(s) = 1 | + 3-s − 2·9-s + 9·11-s − 3·17-s + 3·25-s − 5·27-s + 9·33-s − 12·41-s − 2·43-s − 49-s − 3·51-s + 21·59-s + 22·67-s − 6·73-s + 3·75-s + 81-s + 3·83-s + 21·89-s + 5·97-s − 18·99-s − 3·107-s + 3·113-s + 39·121-s − 12·123-s + 127-s − 2·129-s + 131-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 2.71·11-s − 0.727·17-s + 3/5·25-s − 0.962·27-s + 1.56·33-s − 1.87·41-s − 0.304·43-s − 1/7·49-s − 0.420·51-s + 2.73·59-s + 2.68·67-s − 0.702·73-s + 0.346·75-s + 1/9·81-s + 0.329·83-s + 2.22·89-s + 0.507·97-s − 1.80·99-s − 0.290·107-s + 0.282·113-s + 3.54·121-s − 1.08·123-s + 0.0887·127-s − 0.176·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.521691327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.521691327\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−8.652915932392000060933405063753, −8.311490364141744128562359780738, −7.78563974135050788903398897794, −6.95668089438756207880488624470, −6.74288519507958602330368243749, −6.50898846065493249698552308980, −5.84571853139193445353459255211, −5.25474633169776399073191035800, −4.71849880769310931773950515955, −4.00112457571542125749711622916, −3.68001858646555020846178540741, −3.24832945706715986713608109311, −2.31854290529002361409111266177, −1.79592184954756590545474479487, −0.866744189025422823763620700749,
0.866744189025422823763620700749, 1.79592184954756590545474479487, 2.31854290529002361409111266177, 3.24832945706715986713608109311, 3.68001858646555020846178540741, 4.00112457571542125749711622916, 4.71849880769310931773950515955, 5.25474633169776399073191035800, 5.84571853139193445353459255211, 6.50898846065493249698552308980, 6.74288519507958602330368243749, 6.95668089438756207880488624470, 7.78563974135050788903398897794, 8.311490364141744128562359780738, 8.652915932392000060933405063753
