| L(s) = 1 | + 2·4-s − 4·5-s + 4·9-s − 4·11-s + 3·16-s + 4·19-s − 8·20-s + 2·25-s + 8·29-s − 16·31-s + 8·36-s − 8·41-s − 8·44-s − 16·45-s + 4·49-s + 16·55-s − 8·61-s + 12·64-s − 48·71-s + 8·76-s − 12·80-s − 6·81-s + 40·89-s − 16·95-s − 16·99-s + 4·100-s − 8·101-s + ⋯ |
| L(s) = 1 | + 4-s − 1.78·5-s + 4/3·9-s − 1.20·11-s + 3/4·16-s + 0.917·19-s − 1.78·20-s + 2/5·25-s + 1.48·29-s − 2.87·31-s + 4/3·36-s − 1.24·41-s − 1.20·44-s − 2.38·45-s + 4/7·49-s + 2.15·55-s − 1.02·61-s + 3/2·64-s − 5.69·71-s + 0.917·76-s − 1.34·80-s − 2/3·81-s + 4.23·89-s − 1.64·95-s − 1.60·99-s + 2/5·100-s − 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.482911818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.482911818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 170 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 9046 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 4726 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 25030 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 20726 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 340 T^{2} + 47206 T^{4} - 340 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.07111776638227987228005218262, −7.02163859801641159292072277368, −7.02055093924657424879317981825, −6.38183701279877631732557967864, −6.34868305805270532060359226539, −5.81829463475314533358373380360, −5.78036351731504364789620090153, −5.69057376491909091736527160628, −5.16498019407367959769613959871, −4.96809708116994034877017462531, −4.92828256707623119604802066003, −4.41947365654654955537969906798, −4.18521076800457297353933908422, −4.02481596462036398613628577523, −4.01333435495089806689972868207, −3.28748953732468969721884185918, −3.21762519825422419956337127133, −3.07076017245275117511126261716, −2.97679914715805688795529702647, −2.19218758115201353359208174167, −2.07415943490982078111931949222, −1.65508754891442004233261627156, −1.50531728781544393547938505997, −0.804195951379066507540454243103, −0.31900821042529512003868514157,
0.31900821042529512003868514157, 0.804195951379066507540454243103, 1.50531728781544393547938505997, 1.65508754891442004233261627156, 2.07415943490982078111931949222, 2.19218758115201353359208174167, 2.97679914715805688795529702647, 3.07076017245275117511126261716, 3.21762519825422419956337127133, 3.28748953732468969721884185918, 4.01333435495089806689972868207, 4.02481596462036398613628577523, 4.18521076800457297353933908422, 4.41947365654654955537969906798, 4.92828256707623119604802066003, 4.96809708116994034877017462531, 5.16498019407367959769613959871, 5.69057376491909091736527160628, 5.78036351731504364789620090153, 5.81829463475314533358373380360, 6.34868305805270532060359226539, 6.38183701279877631732557967864, 7.02055093924657424879317981825, 7.02163859801641159292072277368, 7.07111776638227987228005218262
