| L(s) = 1 | − 2·3-s − 2·5-s + 7-s + 3·9-s + 7·11-s − 2·13-s + 4·15-s − 5·17-s + 2·19-s − 2·21-s − 9·23-s + 3·25-s − 4·27-s + 6·29-s − 14·33-s − 2·35-s − 13·37-s + 4·39-s + 41-s − 10·43-s − 6·45-s − 8·47-s − 5·49-s + 10·51-s + 3·53-s − 14·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.377·7-s + 9-s + 2.11·11-s − 0.554·13-s + 1.03·15-s − 1.21·17-s + 0.458·19-s − 0.436·21-s − 1.87·23-s + 3/5·25-s − 0.769·27-s + 1.11·29-s − 2.43·33-s − 0.338·35-s − 2.13·37-s + 0.640·39-s + 0.156·41-s − 1.52·43-s − 0.894·45-s − 1.16·47-s − 5/7·49-s + 1.40·51-s + 0.412·53-s − 1.88·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 13 T + 108 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 144 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 17 T + 206 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - T + 150 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 13 T + 212 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T + 192 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−7.75119508293443106441404280917, −7.51827656602765127593866201870, −6.91466643496912990542109618319, −6.81795617047769813173103427684, −6.43010803814193596654875110228, −6.36805157538887714327708333119, −5.76243808400279733790047628657, −5.25242386947782923041306580521, −4.96680576556264651035993779605, −4.67302326883488512982372469738, −4.19451499187938018385984291632, −3.98212836518085891481431826202, −3.48536006257659077173942147567, −3.32494160183626309016854614940, −2.25125996155115204595125723872, −2.11043362052250107576283366476, −1.31679805159273355004191117877, −1.14599286708228654067965539592, 0, 0,
1.14599286708228654067965539592, 1.31679805159273355004191117877, 2.11043362052250107576283366476, 2.25125996155115204595125723872, 3.32494160183626309016854614940, 3.48536006257659077173942147567, 3.98212836518085891481431826202, 4.19451499187938018385984291632, 4.67302326883488512982372469738, 4.96680576556264651035993779605, 5.25242386947782923041306580521, 5.76243808400279733790047628657, 6.36805157538887714327708333119, 6.43010803814193596654875110228, 6.81795617047769813173103427684, 6.91466643496912990542109618319, 7.51827656602765127593866201870, 7.75119508293443106441404280917
