| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 9-s + 12·13-s + 5·16-s − 8·17-s + 2·18-s + 10·19-s − 24·26-s − 6·32-s + 16·34-s − 3·36-s − 20·38-s − 18·43-s + 14·47-s + 5·49-s + 36·52-s − 18·53-s + 7·64-s − 26·67-s − 24·68-s + 4·72-s + 30·76-s + 81-s + 12·83-s + 36·86-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 1/3·9-s + 3.32·13-s + 5/4·16-s − 1.94·17-s + 0.471·18-s + 2.29·19-s − 4.70·26-s − 1.06·32-s + 2.74·34-s − 1/2·36-s − 3.24·38-s − 2.74·43-s + 2.04·47-s + 5/7·49-s + 4.99·52-s − 2.47·53-s + 7/8·64-s − 3.17·67-s − 2.91·68-s + 0.471·72-s + 3.44·76-s + 1/9·81-s + 1.31·83-s + 3.88·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.281610833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.281610833\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.939356424860346497372138198987, −8.750097111912997300861064047696, −8.460748441561197525015952767386, −8.171950665363564383955980946475, −7.55534081073156664536225998046, −7.36494255444038018413804892445, −6.85995922850034180564809677020, −6.37423533405022419345702480257, −6.15829847009037121729998324506, −5.86812092560260338695919931056, −5.37775793615889047836584081626, −4.72409965906794031891673785284, −4.29941942992926682234537392217, −3.54814187292834819690867394929, −3.32781846637359992257630067843, −2.99786158009943644466650984735, −2.15001075824131315984038297081, −1.55356393873671629363159143849, −1.26208608543052845014754216395, −0.52213276432786156045869401896,
0.52213276432786156045869401896, 1.26208608543052845014754216395, 1.55356393873671629363159143849, 2.15001075824131315984038297081, 2.99786158009943644466650984735, 3.32781846637359992257630067843, 3.54814187292834819690867394929, 4.29941942992926682234537392217, 4.72409965906794031891673785284, 5.37775793615889047836584081626, 5.86812092560260338695919931056, 6.15829847009037121729998324506, 6.37423533405022419345702480257, 6.85995922850034180564809677020, 7.36494255444038018413804892445, 7.55534081073156664536225998046, 8.171950665363564383955980946475, 8.460748441561197525015952767386, 8.750097111912997300861064047696, 8.939356424860346497372138198987
