| L(s) = 1 | + 4-s + 2·5-s + 2·7-s − 8·11-s − 3·16-s + 10·17-s + 10·19-s + 2·20-s − 2·23-s − 2·25-s + 2·28-s − 4·31-s + 4·35-s + 4·41-s + 2·43-s − 8·44-s + 8·47-s − 6·49-s + 6·53-s − 16·55-s − 8·59-s − 7·64-s + 6·67-s + 10·68-s + 16·71-s − 4·73-s + 10·76-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.894·5-s + 0.755·7-s − 2.41·11-s − 3/4·16-s + 2.42·17-s + 2.29·19-s + 0.447·20-s − 0.417·23-s − 2/5·25-s + 0.377·28-s − 0.718·31-s + 0.676·35-s + 0.624·41-s + 0.304·43-s − 1.20·44-s + 1.16·47-s − 6/7·49-s + 0.824·53-s − 2.15·55-s − 1.04·59-s − 7/8·64-s + 0.733·67-s + 1.21·68-s + 1.89·71-s − 0.468·73-s + 1.14·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42849 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42849 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.765192605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.765192605\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 174 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.41635686132260183619352698019, −12.26819729161161278646565244989, −11.70634698476176216317390232283, −10.98870331069602887125074447346, −10.83265173232447580992191552433, −10.14881173928693498675562834367, −9.622293403792038343965844034665, −9.614593930167483088583521843275, −8.569419856824371148104097405995, −7.83146599695803646210358425141, −7.64395047814074730330078373815, −7.40368496525374545864537169215, −6.35969305762392117272338848679, −5.50476019734614739065412695050, −5.43198506095900549009701604360, −5.06796354574754375643604479623, −3.82089084173819696868734908335, −2.91408960709706569416398030595, −2.43350663926757041321316798946, −1.36694706165124286307138591500,
1.36694706165124286307138591500, 2.43350663926757041321316798946, 2.91408960709706569416398030595, 3.82089084173819696868734908335, 5.06796354574754375643604479623, 5.43198506095900549009701604360, 5.50476019734614739065412695050, 6.35969305762392117272338848679, 7.40368496525374545864537169215, 7.64395047814074730330078373815, 7.83146599695803646210358425141, 8.569419856824371148104097405995, 9.614593930167483088583521843275, 9.622293403792038343965844034665, 10.14881173928693498675562834367, 10.83265173232447580992191552433, 10.98870331069602887125074447346, 11.70634698476176216317390232283, 12.26819729161161278646565244989, 12.41635686132260183619352698019
