| L(s) = 1 | + 6·5-s − 9-s − 2·13-s − 6·17-s + 17·25-s + 20·29-s + 6·37-s − 6·45-s − 9·49-s + 8·53-s − 12·65-s + 28·73-s − 8·81-s − 36·85-s − 20·89-s − 4·97-s − 24·101-s + 22·109-s − 28·113-s + 2·117-s − 2·121-s + 18·125-s + 127-s + 131-s + 137-s + 139-s + 120·145-s + ⋯ |
| L(s) = 1 | + 2.68·5-s − 1/3·9-s − 0.554·13-s − 1.45·17-s + 17/5·25-s + 3.71·29-s + 0.986·37-s − 0.894·45-s − 9/7·49-s + 1.09·53-s − 1.48·65-s + 3.27·73-s − 8/9·81-s − 3.90·85-s − 2.11·89-s − 0.406·97-s − 2.38·101-s + 2.10·109-s − 2.63·113-s + 0.184·117-s − 0.181·121-s + 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 9.96·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.650069237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.650069237\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−11.27110648180870015477247594486, −10.90615349797839564088435717397, −10.26361615459714924439044485613, −10.15917061080704108074497661375, −9.583026452911920925962200783250, −9.392098843380701832603579291700, −8.826516368022426000035757705537, −8.314822912965428696565169918059, −7.964788732557451643835501218434, −6.81927019276186980443433613696, −6.53859565101617360290440907337, −6.49409899418294038372661757501, −5.59339610543323639627859383173, −5.41266641732841925902519708962, −4.68488467222293078851897360358, −4.28579371422244664849394136141, −2.95529952262336145499629650859, −2.54111305152544911453321665086, −2.09393659444886036309468602636, −1.15759207088729262449011784997,
1.15759207088729262449011784997, 2.09393659444886036309468602636, 2.54111305152544911453321665086, 2.95529952262336145499629650859, 4.28579371422244664849394136141, 4.68488467222293078851897360358, 5.41266641732841925902519708962, 5.59339610543323639627859383173, 6.49409899418294038372661757501, 6.53859565101617360290440907337, 6.81927019276186980443433613696, 7.964788732557451643835501218434, 8.314822912965428696565169918059, 8.826516368022426000035757705537, 9.392098843380701832603579291700, 9.583026452911920925962200783250, 10.15917061080704108074497661375, 10.26361615459714924439044485613, 10.90615349797839564088435717397, 11.27110648180870015477247594486
