| L(s) = 1 | − 9-s − 2·19-s + 2·31-s + 49-s − 2·61-s + 4·79-s + 81-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 2·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 9-s − 2·19-s + 2·31-s + 49-s − 2·61-s + 4·79-s + 81-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 2·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8531457649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8531457649\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$ | \( ( 1 - T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
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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
−10.16692126896792320217436702249, −9.712988721814415946608443269753, −9.324223472328552274411279417898, −8.760434694483442977948695514813, −8.637800860934582289020213754342, −8.076622587861965813696973815329, −7.915532844020275280223287750422, −7.28822144623843650417256240064, −6.66021730851117044885065126857, −6.44497921649453797621235043873, −5.99161771659389753301739223094, −5.65924444420486887252769112857, −4.96767717499576217254459612318, −4.48280741826464614963215358653, −4.28823171968519359752415878042, −3.36691859714621788828279753969, −3.14018841927764738210195200695, −2.22942052793984426763547218566, −2.10863678667909022249764221974, −0.817156158108819163320116772205,
0.817156158108819163320116772205, 2.10863678667909022249764221974, 2.22942052793984426763547218566, 3.14018841927764738210195200695, 3.36691859714621788828279753969, 4.28823171968519359752415878042, 4.48280741826464614963215358653, 4.96767717499576217254459612318, 5.65924444420486887252769112857, 5.99161771659389753301739223094, 6.44497921649453797621235043873, 6.66021730851117044885065126857, 7.28822144623843650417256240064, 7.915532844020275280223287750422, 8.076622587861965813696973815329, 8.637800860934582289020213754342, 8.760434694483442977948695514813, 9.324223472328552274411279417898, 9.712988721814415946608443269753, 10.16692126896792320217436702249
