| L(s) = 1 | + 4·11-s − 4·19-s + 16·29-s − 16·31-s + 12·41-s + 10·49-s − 4·59-s − 20·61-s + 40·71-s − 16·79-s + 2·81-s − 8·89-s − 40·101-s + 60·109-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 42·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 0.917·19-s + 2.97·29-s − 2.87·31-s + 1.87·41-s + 10/7·49-s − 0.520·59-s − 2.56·61-s + 4.74·71-s − 1.80·79-s + 2/9·81-s − 0.847·89-s − 3.98·101-s + 5.74·109-s − 3.09·121-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 + 3.23·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.181812209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.181812209\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 - 10 T^{2} + 43 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 24 T^{2} + 1262 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 67 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 494 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 15598 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 5302 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 202 T^{2} + 18859 T^{4} - 202 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 8254 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.80746839239138115610532103775, −5.71339902458204320830556154845, −5.43618571507306981409759320994, −5.32588731330798962974952497215, −5.23340430667200712920764933102, −4.69972495676999912759056613188, −4.63355156205046568612357614609, −4.57723030009085768470473059861, −4.34887953236486668901650709436, −3.89867415332916577072895792677, −3.87829425719174037918239165735, −3.76928302661789078441648431605, −3.69119547453675912203272561849, −3.12389559337084299398196648694, −2.95971726648781881940859922863, −2.80007157908086279738788731897, −2.61261581539510535648244937039, −2.29679689920389995367326545886, −2.05923097985491123177103830134, −1.79175041630982347049259360433, −1.53409519405692393840856577057, −1.25716067374894913814630004074, −0.861547556275380727622899041899, −0.799973499718488600341583529512, −0.14334852157426483231730937855,
0.14334852157426483231730937855, 0.799973499718488600341583529512, 0.861547556275380727622899041899, 1.25716067374894913814630004074, 1.53409519405692393840856577057, 1.79175041630982347049259360433, 2.05923097985491123177103830134, 2.29679689920389995367326545886, 2.61261581539510535648244937039, 2.80007157908086279738788731897, 2.95971726648781881940859922863, 3.12389559337084299398196648694, 3.69119547453675912203272561849, 3.76928302661789078441648431605, 3.87829425719174037918239165735, 3.89867415332916577072895792677, 4.34887953236486668901650709436, 4.57723030009085768470473059861, 4.63355156205046568612357614609, 4.69972495676999912759056613188, 5.23340430667200712920764933102, 5.32588731330798962974952497215, 5.43618571507306981409759320994, 5.71339902458204320830556154845, 5.80746839239138115610532103775
