| L(s) = 1 | + 3-s + 3·5-s − 4·9-s + 3·15-s − 2·17-s + 4·19-s − 14·23-s − 2·25-s − 6·27-s + 4·29-s + 7·31-s + 8·37-s + 9·41-s − 9·43-s − 12·45-s + 18·47-s − 2·51-s + 5·53-s + 4·57-s + 8·59-s − 13·61-s + 11·67-s − 14·69-s + 3·73-s − 2·75-s − 6·79-s + 8·81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 4/3·9-s + 0.774·15-s − 0.485·17-s + 0.917·19-s − 2.91·23-s − 2/5·25-s − 1.15·27-s + 0.742·29-s + 1.25·31-s + 1.31·37-s + 1.40·41-s − 1.37·43-s − 1.78·45-s + 2.62·47-s − 0.280·51-s + 0.686·53-s + 0.529·57-s + 1.04·59-s − 1.66·61-s + 1.34·67-s − 1.68·69-s + 0.351·73-s − 0.230·75-s − 0.675·79-s + 8/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44408896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44408896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.848108731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.848108731\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 14 T + 90 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 75 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 81 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 163 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 11 T + 133 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 242 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 158 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T - 17 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−8.194236498817811408310884688340, −7.941324683884433257266337829855, −7.44880120488527896452976758793, −7.36171742474818568335847441292, −6.47225672625500998585027008920, −6.30169116369892121265963741874, −6.01198927418271405001118508630, −5.89111199141954782635656305458, −5.38450113866789782789353591690, −5.15733433623260229622983960574, −4.35958957616347710224273411126, −4.33924656998569319803950923664, −3.65502559918208405714508557069, −3.43915054770371247339421615984, −2.63054236298960805733130400838, −2.59365546078979002359444578442, −2.06458656299424636767006347998, −1.94919612182651566776652957879, −0.989379091681628955044441504407, −0.50187999603879747533974664013,
0.50187999603879747533974664013, 0.989379091681628955044441504407, 1.94919612182651566776652957879, 2.06458656299424636767006347998, 2.59365546078979002359444578442, 2.63054236298960805733130400838, 3.43915054770371247339421615984, 3.65502559918208405714508557069, 4.33924656998569319803950923664, 4.35958957616347710224273411126, 5.15733433623260229622983960574, 5.38450113866789782789353591690, 5.89111199141954782635656305458, 6.01198927418271405001118508630, 6.30169116369892121265963741874, 6.47225672625500998585027008920, 7.36171742474818568335847441292, 7.44880120488527896452976758793, 7.941324683884433257266337829855, 8.194236498817811408310884688340
