| L(s) = 1 | + 2·5-s − 6·7-s − 2·13-s − 2·19-s + 4·23-s − 2·25-s + 8·29-s − 14·31-s − 12·35-s + 14·41-s − 4·43-s + 8·47-s + 18·49-s + 8·53-s + 4·59-s + 16·61-s − 4·65-s + 2·67-s + 20·71-s + 12·83-s + 18·89-s + 12·91-s − 4·95-s + 24·97-s − 20·101-s − 4·103-s + 4·107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2.26·7-s − 0.554·13-s − 0.458·19-s + 0.834·23-s − 2/5·25-s + 1.48·29-s − 2.51·31-s − 2.02·35-s + 2.18·41-s − 0.609·43-s + 1.16·47-s + 18/7·49-s + 1.09·53-s + 0.520·59-s + 2.04·61-s − 0.496·65-s + 0.244·67-s + 2.37·71-s + 1.31·83-s + 1.90·89-s + 1.25·91-s − 0.410·95-s + 2.43·97-s − 1.99·101-s − 0.394·103-s + 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.075156148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.075156148\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 14 T + 106 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 318 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−8.753547240708029640517418113939, −8.577704118718793588584107541703, −7.74311154122299280911435733013, −7.64544119778910187158050074634, −7.00982183408113719202093453382, −6.84828542880571968956839474654, −6.47627095113072779934965590084, −6.18108982754412651958455709585, −5.74051930310109236804608742115, −5.40943589360785531880833993398, −5.12047833333735106290559814981, −4.48492848311660777217867110703, −3.80777487696000384217234411151, −3.76585321555109278045486092470, −3.21469886391956840938928740929, −2.69086027561105803285804237516, −2.21505151502219968764515174520, −2.08151906722619159786089501973, −0.883049360357620370781943413437, −0.52893973460632317371565030238,
0.52893973460632317371565030238, 0.883049360357620370781943413437, 2.08151906722619159786089501973, 2.21505151502219968764515174520, 2.69086027561105803285804237516, 3.21469886391956840938928740929, 3.76585321555109278045486092470, 3.80777487696000384217234411151, 4.48492848311660777217867110703, 5.12047833333735106290559814981, 5.40943589360785531880833993398, 5.74051930310109236804608742115, 6.18108982754412651958455709585, 6.47627095113072779934965590084, 6.84828542880571968956839474654, 7.00982183408113719202093453382, 7.64544119778910187158050074634, 7.74311154122299280911435733013, 8.577704118718793588584107541703, 8.753547240708029640517418113939
