| L(s) = 1 | + 8·7-s + 9-s − 4·17-s − 16·23-s + 25-s + 20·41-s + 16·47-s + 34·49-s + 8·63-s − 28·73-s + 32·79-s + 81-s + 4·89-s + 4·97-s + 8·103-s + 12·113-s − 32·119-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s − 128·161-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 1/3·9-s − 0.970·17-s − 3.33·23-s + 1/5·25-s + 3.12·41-s + 2.33·47-s + 34/7·49-s + 1.00·63-s − 3.27·73-s + 3.60·79-s + 1/9·81-s + 0.423·89-s + 0.406·97-s + 0.788·103-s + 1.12·113-s − 2.93·119-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.323·153-s + 0.0798·157-s − 10.0·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.191749975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.191749975\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.138788409882082860623313179234, −9.082691646086291194533627292052, −8.338450200049268291048738447687, −7.929044973464791963060046214031, −7.64493796343622540106320647343, −7.34594365176835478558310775458, −6.25442273107448540439252397130, −5.92809906508794084871962623137, −5.28897888504504330389281711769, −4.67467612086321652791764027956, −4.13728859524865050162716547657, −4.05477376880357436694530570270, −2.25919867640796924053383456153, −2.19180563382997996757520036314, −1.20192952395638838673519576448,
1.20192952395638838673519576448, 2.19180563382997996757520036314, 2.25919867640796924053383456153, 4.05477376880357436694530570270, 4.13728859524865050162716547657, 4.67467612086321652791764027956, 5.28897888504504330389281711769, 5.92809906508794084871962623137, 6.25442273107448540439252397130, 7.34594365176835478558310775458, 7.64493796343622540106320647343, 7.929044973464791963060046214031, 8.338450200049268291048738447687, 9.082691646086291194533627292052, 9.138788409882082860623313179234
