| L(s) = 1 | − 10·9-s + 40·13-s − 20·17-s − 38·25-s − 160·29-s − 46·49-s + 40·53-s + 160·61-s − 87·81-s + 208·101-s + 40·113-s − 400·117-s − 100·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 200·153-s + 157-s + 163-s + 167-s + 862·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.11·9-s + 3.07·13-s − 1.17·17-s − 1.51·25-s − 5.51·29-s − 0.938·49-s + 0.754·53-s + 2.62·61-s − 1.07·81-s + 2.05·101-s + 0.353·113-s − 3.41·117-s − 0.826·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 1.30·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 5.10·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.267547026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.267547026\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 + 5 T^{2} + p^{4} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 710 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 p T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 1490 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1013 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 1835 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 1657 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 2630 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 1822 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 9215 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 3758 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 8218 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13478 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 8942 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 8882 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.12602520888559376802844701154, −6.88475691733814945617030800399, −6.63623073584368899095370878068, −6.44232578129918051875450237927, −5.95172124297524132111167581998, −5.88645589491455845466279872257, −5.79210771237298611931130248238, −5.74981641944274741796422330244, −5.29858153473202286045862479321, −5.17175397934008883980612242611, −4.82506902574281737094248580334, −4.24554752547903162917047911807, −4.05926862552703457064648489796, −4.00401524844037044231925040550, −3.72709319612568352332784464540, −3.43735675201342973265369876563, −3.32519894900346866695385289130, −2.99075512472621930911330746437, −2.25771636584362839126097995097, −2.23287153949932623455361191369, −1.94118205391733956502527554885, −1.52227262690839896806596496050, −1.33385465951921127242116032953, −0.46199678033481057364057735260, −0.37264892199664266296029216095,
0.37264892199664266296029216095, 0.46199678033481057364057735260, 1.33385465951921127242116032953, 1.52227262690839896806596496050, 1.94118205391733956502527554885, 2.23287153949932623455361191369, 2.25771636584362839126097995097, 2.99075512472621930911330746437, 3.32519894900346866695385289130, 3.43735675201342973265369876563, 3.72709319612568352332784464540, 4.00401524844037044231925040550, 4.05926862552703457064648489796, 4.24554752547903162917047911807, 4.82506902574281737094248580334, 5.17175397934008883980612242611, 5.29858153473202286045862479321, 5.74981641944274741796422330244, 5.79210771237298611931130248238, 5.88645589491455845466279872257, 5.95172124297524132111167581998, 6.44232578129918051875450237927, 6.63623073584368899095370878068, 6.88475691733814945617030800399, 7.12602520888559376802844701154
