| L(s) = 1 | − 243·9-s − 2.40e3·13-s + 6.25e3·25-s + 3.31e4·37-s + 2.20e4·49-s − 7.72e4·61-s + 2.90e3·73-s + 5.90e4·81-s + 2.68e5·97-s − 2.28e5·109-s + 5.84e5·117-s − 3.22e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.59e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 9-s − 3.94·13-s + 2·25-s + 3.97·37-s + 1.31·49-s − 2.65·61-s + 0.0636·73-s + 81-s + 2.90·97-s − 1.84·109-s + 3.94·117-s − 2·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 9.67·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.115717297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.115717297\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{5} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 236 T + p^{5} T^{2} )( 1 + 236 T + p^{5} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 1202 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 1432 T + p^{5} T^{2} )( 1 + 1432 T + p^{5} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10324 T + p^{5} T^{2} )( 1 + 10324 T + p^{5} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 16550 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 3352 T + p^{5} T^{2} )( 1 + 3352 T + p^{5} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 38626 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 35536 T + p^{5} T^{2} )( 1 + 35536 T + p^{5} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1450 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 100564 T + p^{5} T^{2} )( 1 + 100564 T + p^{5} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 134386 T + p^{5} T^{2} )^{2} \) |
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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
−14.88833208845833776686730933753, −14.54987820751536153950059612806, −14.03699098206157437608019050711, −13.02435106977880487577777503141, −12.60489670257376498889653323020, −12.00894012259824749626309218686, −11.62697941999381711530319783869, −10.73294575317049869647731396624, −10.16702957579116635747158192743, −9.325957207235366741403073429107, −9.202446376916645672987468336133, −7.891029212429343711132222885741, −7.58625326679362375573330017890, −6.81550699366384216420948206783, −5.86379323120899748946715646826, −4.90379891855357016209382653030, −4.57919247105348997594189761510, −2.70826022295042964647823495122, −2.58921108622822011621847214912, −0.52985752654061047898380275980,
0.52985752654061047898380275980, 2.58921108622822011621847214912, 2.70826022295042964647823495122, 4.57919247105348997594189761510, 4.90379891855357016209382653030, 5.86379323120899748946715646826, 6.81550699366384216420948206783, 7.58625326679362375573330017890, 7.891029212429343711132222885741, 9.202446376916645672987468336133, 9.325957207235366741403073429107, 10.16702957579116635747158192743, 10.73294575317049869647731396624, 11.62697941999381711530319783869, 12.00894012259824749626309218686, 12.60489670257376498889653323020, 13.02435106977880487577777503141, 14.03699098206157437608019050711, 14.54987820751536153950059612806, 14.88833208845833776686730933753
