| L(s) = 1 | − 192·13-s + 2.49e3·25-s − 5.17e3·37-s + 4.22e3·49-s − 1.36e4·61-s + 3.60e4·73-s + 4.03e4·97-s + 3.78e4·109-s + 3.09e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.12e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 1.13·13-s + 3.99·25-s − 3.78·37-s + 1.75·49-s − 3.66·61-s + 6.77·73-s + 4.28·97-s + 3.18·109-s + 2.11·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 3.19·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.797012821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.797012821\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 1248 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 2110 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 15458 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 48 T + p^{4} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 162624 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 150050 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 214082 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 753312 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1784834 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1294 T + p^{4} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 4745664 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 2165090 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 3525502 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 6144480 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 4272866 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 3410 T + p^{4} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 17050274 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9005762 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9024 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 74851970 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 51742174 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 106028160 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10080 T + p^{4} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.06166582012755971928421306208, −6.94364935794898079372322973123, −6.54238092101057572233374403446, −6.50422842035854052359955241815, −6.30152200159557313989007711624, −5.96054206201405225216861400309, −5.44135365582639750700146141476, −5.21459401435408158959110211067, −5.20629835339480768887768142335, −4.98390398592847309588920655984, −4.58124478096700401233516258820, −4.53588880789103958250334255877, −4.21085430481760443627733731119, −3.61171825928589859880705044870, −3.36299368921337163124023029187, −3.21962354935771830678031434293, −3.16311381823972345541706287120, −2.63188384635991837874834269895, −2.12757186689696359293029007561, −2.05660524268619477760507451577, −1.84389498373972346112778943912, −1.21406404731698329824711990375, −0.74437379598995550289476029013, −0.70621752834167332121484512154, −0.33674690645780207444027684965,
0.33674690645780207444027684965, 0.70621752834167332121484512154, 0.74437379598995550289476029013, 1.21406404731698329824711990375, 1.84389498373972346112778943912, 2.05660524268619477760507451577, 2.12757186689696359293029007561, 2.63188384635991837874834269895, 3.16311381823972345541706287120, 3.21962354935771830678031434293, 3.36299368921337163124023029187, 3.61171825928589859880705044870, 4.21085430481760443627733731119, 4.53588880789103958250334255877, 4.58124478096700401233516258820, 4.98390398592847309588920655984, 5.20629835339480768887768142335, 5.21459401435408158959110211067, 5.44135365582639750700146141476, 5.96054206201405225216861400309, 6.30152200159557313989007711624, 6.50422842035854052359955241815, 6.54238092101057572233374403446, 6.94364935794898079372322973123, 7.06166582012755971928421306208
