L(s) = 1 | − 12·13-s + 4·17-s − 20·37-s − 4·53-s + 16·61-s + 12·73-s − 81-s + 12·97-s + 72·101-s − 44·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | − 3.32·13-s + 0.970·17-s − 3.28·37-s − 0.549·53-s + 2.04·61-s + 1.40·73-s − 1/9·81-s + 1.21·97-s + 7.16·101-s − 4.13·113-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1601650980\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1601650980\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 1202 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 1666 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 8722 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.01077617031745985239779039422, −6.80314370290963417859962625868, −6.64987701930415734535918200463, −6.20250003676782334101399318827, −6.08245503354473245912528622181, −6.05530462323691858004681362040, −5.39516422145205998460850088427, −5.26328172477102018476667521457, −5.15465300860496654286150259793, −4.89674405062634080836484397402, −4.85959361786151824557652541883, −4.79642445434659936435259054665, −3.99654835336251417459405434039, −3.92988506579200769464833640712, −3.82695376758861611073125842380, −3.43365806537013566984352240743, −3.07801849141329416927824617837, −2.95930554230974642069094052993, −2.54464444154210464650021074120, −2.16989105614327311718151528191, −2.14363232076853599274413629038, −1.82988116244737665243717049320, −1.23704812270115128389863172673, −0.836640434130778066386649387967, −0.095019547456513772752380996130,
0.095019547456513772752380996130, 0.836640434130778066386649387967, 1.23704812270115128389863172673, 1.82988116244737665243717049320, 2.14363232076853599274413629038, 2.16989105614327311718151528191, 2.54464444154210464650021074120, 2.95930554230974642069094052993, 3.07801849141329416927824617837, 3.43365806537013566984352240743, 3.82695376758861611073125842380, 3.92988506579200769464833640712, 3.99654835336251417459405434039, 4.79642445434659936435259054665, 4.85959361786151824557652541883, 4.89674405062634080836484397402, 5.15465300860496654286150259793, 5.26328172477102018476667521457, 5.39516422145205998460850088427, 6.05530462323691858004681362040, 6.08245503354473245912528622181, 6.20250003676782334101399318827, 6.64987701930415734535918200463, 6.80314370290963417859962625868, 7.01077617031745985239779039422