| L(s) = 1 | − 2·3-s + 2·5-s − 8·7-s + 2·13-s − 4·15-s + 2·17-s + 16·21-s − 6·23-s − 4·25-s + 2·27-s + 4·29-s − 4·31-s − 16·35-s − 6·37-s − 4·39-s − 6·41-s + 4·43-s + 12·47-s + 34·49-s − 4·51-s + 16·53-s − 8·59-s − 8·61-s + 4·65-s + 16·67-s + 12·69-s − 16·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 3.02·7-s + 0.554·13-s − 1.03·15-s + 0.485·17-s + 3.49·21-s − 1.25·23-s − 4/5·25-s + 0.384·27-s + 0.742·29-s − 0.718·31-s − 2.70·35-s − 0.986·37-s − 0.640·39-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 34/7·49-s − 0.560·51-s + 2.19·53-s − 1.04·59-s − 1.02·61-s + 0.496·65-s + 1.95·67-s + 1.44·69-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12503296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12503296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 56 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 144 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 84 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 230 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 22 T + 312 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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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
−8.460538815456688328481205993731, −8.037194099811228383924211927188, −7.36020473338281624412852596890, −7.19948991277741604452250996231, −6.63003599435402986145391704695, −6.46552867412155763163218372107, −6.03434375398785144033729454088, −5.85761148467138629866953275572, −5.52804108094712736121310347832, −5.41311672520583685002841284676, −4.49865684171640042626891211589, −4.04048850200020776176405760142, −3.52927440589023910156708939722, −3.45792161004300489504810991087, −2.65368177442337389097960934072, −2.49397585031502342920046888935, −1.69798210296866562437122604327, −0.940013542616972185456784318435, 0, 0,
0.940013542616972185456784318435, 1.69798210296866562437122604327, 2.49397585031502342920046888935, 2.65368177442337389097960934072, 3.45792161004300489504810991087, 3.52927440589023910156708939722, 4.04048850200020776176405760142, 4.49865684171640042626891211589, 5.41311672520583685002841284676, 5.52804108094712736121310347832, 5.85761148467138629866953275572, 6.03434375398785144033729454088, 6.46552867412155763163218372107, 6.63003599435402986145391704695, 7.19948991277741604452250996231, 7.36020473338281624412852596890, 8.037194099811228383924211927188, 8.460538815456688328481205993731
