| L(s) = 1 | + 4·2-s + 8·4-s + 8·8-s − 4·16-s − 20·29-s − 32·32-s − 12·37-s + 10·49-s − 16·53-s − 80·58-s − 64·64-s − 48·74-s − 18·81-s + 40·98-s − 64·106-s − 60·109-s + 44·113-s − 160·116-s − 6·121-s + 127-s − 64·128-s + 131-s + 137-s + 139-s − 96·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 2.82·8-s − 16-s − 3.71·29-s − 5.65·32-s − 1.97·37-s + 10/7·49-s − 2.19·53-s − 10.5·58-s − 8·64-s − 5.57·74-s − 2·81-s + 4.04·98-s − 6.21·106-s − 5.74·109-s + 4.13·113-s − 14.8·116-s − 0.545·121-s + 0.0887·127-s − 5.65·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 7.89·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.132837469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.132837469\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 125 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 117 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 77 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 172 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.33603934988985543194578476251, −7.15890708456581685775623366036, −7.10453254970657288572452762871, −6.52957381659368619202951981033, −6.49215603335186274682406037013, −6.20982657808185776701927523402, −6.00823123341572054510092192746, −5.75982599054109470485272139701, −5.38005185479948006116091359946, −5.28466109350992982297749775141, −5.21951890903477043774803595325, −5.03332184204284970418156973161, −4.45691612973383970469121465518, −4.34246573874357043484016938887, −3.97302133457048139043460613086, −3.89539643778979726450819971188, −3.71281574786044364648898328731, −3.25462283823779950129963305013, −3.10441690704221457358367747587, −2.84154108215813445422199488833, −2.40132893601016522308064971155, −2.07287317658215946414262142072, −1.71604336752395738307051425510, −1.45764750146481887134915823969, −0.20929946574667686533003357444,
0.20929946574667686533003357444, 1.45764750146481887134915823969, 1.71604336752395738307051425510, 2.07287317658215946414262142072, 2.40132893601016522308064971155, 2.84154108215813445422199488833, 3.10441690704221457358367747587, 3.25462283823779950129963305013, 3.71281574786044364648898328731, 3.89539643778979726450819971188, 3.97302133457048139043460613086, 4.34246573874357043484016938887, 4.45691612973383970469121465518, 5.03332184204284970418156973161, 5.21951890903477043774803595325, 5.28466109350992982297749775141, 5.38005185479948006116091359946, 5.75982599054109470485272139701, 6.00823123341572054510092192746, 6.20982657808185776701927523402, 6.49215603335186274682406037013, 6.52957381659368619202951981033, 7.10453254970657288572452762871, 7.15890708456581685775623366036, 7.33603934988985543194578476251
