| L(s) = 1 | − 2·3-s − 2·4-s + 3·5-s + 9-s + 4·12-s − 6·15-s − 6·20-s − 3·23-s + 4·25-s + 4·27-s − 2·31-s − 2·36-s − 18·37-s + 3·45-s − 21·47-s + 2·49-s + 18·53-s − 15·59-s + 12·60-s + 8·64-s + 6·69-s − 21·71-s − 8·75-s − 11·81-s − 9·89-s + 6·92-s + 4·93-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 1.34·5-s + 1/3·9-s + 1.15·12-s − 1.54·15-s − 1.34·20-s − 0.625·23-s + 4/5·25-s + 0.769·27-s − 0.359·31-s − 1/3·36-s − 2.95·37-s + 0.447·45-s − 3.06·47-s + 2/7·49-s + 2.47·53-s − 1.95·59-s + 1.54·60-s + 64-s + 0.722·69-s − 2.49·71-s − 0.923·75-s − 1.22·81-s − 0.953·89-s + 0.625·92-s + 0.414·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3248454636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3248454636\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−7.30824845984976924238990703509, −6.93275094325729312340701527268, −6.62999563558761094718762867470, −6.15600627177212845264506692506, −5.69123742621604292966783911979, −5.53185949754034281137058368951, −4.98374361647872624899648170312, −4.80888308439694567367942689707, −4.24491109938982553301303420029, −3.64729513921333045933162350227, −3.14670017183830822086441793621, −2.46331553769077827732073358538, −1.73580582125540547611989809817, −1.39383683977412058707685019539, −0.23377236844491829273595134830,
0.23377236844491829273595134830, 1.39383683977412058707685019539, 1.73580582125540547611989809817, 2.46331553769077827732073358538, 3.14670017183830822086441793621, 3.64729513921333045933162350227, 4.24491109938982553301303420029, 4.80888308439694567367942689707, 4.98374361647872624899648170312, 5.53185949754034281137058368951, 5.69123742621604292966783911979, 6.15600627177212845264506692506, 6.62999563558761094718762867470, 6.93275094325729312340701527268, 7.30824845984976924238990703509
