L(s) = 1 | + 4-s − 6·9-s − 4·11-s − 16·19-s − 12·29-s − 8·31-s − 6·36-s + 6·41-s − 4·44-s − 4·49-s + 18·59-s + 24·61-s − 64-s − 12·71-s − 16·76-s − 28·79-s + 9·81-s − 2·89-s + 24·99-s − 20·101-s + 20·109-s − 12·116-s − 34·121-s − 8·124-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 2·9-s − 1.20·11-s − 3.67·19-s − 2.22·29-s − 1.43·31-s − 36-s + 0.937·41-s − 0.603·44-s − 4/7·49-s + 2.34·59-s + 3.07·61-s − 1/8·64-s − 1.42·71-s − 1.83·76-s − 3.15·79-s + 81-s − 0.211·89-s + 2.41·99-s − 1.99·101-s + 1.91·109-s − 1.11·116-s − 3.09·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05346192357\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05346192357\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 58 T^{2} + 1515 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 91 T^{2} + 3792 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 193 T^{2} + 27840 T^{4} + 193 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.12654592791856077799503944032, −7.12132474426389147860248536190, −6.55064257551126515120032214017, −6.50649615988930135671130402541, −6.41234531765191042081676305317, −6.07938407972285451453570621602, −5.65675915745214051743828982256, −5.58397859904371352651364441133, −5.43789643344384683988432976920, −5.31862464841609411557547049804, −5.10507947614669028854543338447, −4.43576366485514292525272644486, −4.37760433405627181451675830326, −3.97374035015413308731977684473, −3.89096328087104399176111917729, −3.79965129387850852672553759948, −3.13530812597439691464877968150, −2.83121471976045221629937198561, −2.64423249354430178508023421281, −2.60879626285126772929398887980, −1.94558489661528996160772268840, −1.94097361328050071675848570483, −1.69702765497134324801395264333, −0.66334955259857418275040190240, −0.06902544753495799583094490710,
0.06902544753495799583094490710, 0.66334955259857418275040190240, 1.69702765497134324801395264333, 1.94097361328050071675848570483, 1.94558489661528996160772268840, 2.60879626285126772929398887980, 2.64423249354430178508023421281, 2.83121471976045221629937198561, 3.13530812597439691464877968150, 3.79965129387850852672553759948, 3.89096328087104399176111917729, 3.97374035015413308731977684473, 4.37760433405627181451675830326, 4.43576366485514292525272644486, 5.10507947614669028854543338447, 5.31862464841609411557547049804, 5.43789643344384683988432976920, 5.58397859904371352651364441133, 5.65675915745214051743828982256, 6.07938407972285451453570621602, 6.41234531765191042081676305317, 6.50649615988930135671130402541, 6.55064257551126515120032214017, 7.12132474426389147860248536190, 7.12654592791856077799503944032