| L(s) = 1 | + 6·9-s − 4·19-s − 8·25-s + 12·29-s − 8·31-s + 6·49-s − 12·59-s + 40·61-s + 48·71-s + 8·79-s + 17·81-s + 24·101-s + 4·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·169-s − 24·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2·9-s − 0.917·19-s − 8/5·25-s + 2.22·29-s − 1.43·31-s + 6/7·49-s − 1.56·59-s + 5.12·61-s + 5.69·71-s + 0.900·79-s + 17/9·81-s + 2.38·101-s + 0.383·109-s − 3.63·121-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 + 1.38·169-s − 1.83·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.001029654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.001029654\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 6 T^{2} + 5 p T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 18 T^{2} + 131 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T^{2} - 733 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 2130 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 3651 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 70 T^{2} + 4371 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 24 T + 4 p T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 130 T^{2} + 12291 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_4$ | \( ( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6774 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 252 T^{2} + 30086 T^{4} - 252 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.70060184641792138131439715967, −6.37148812261193787453585518580, −6.32502142692066894376224366107, −6.26713134262306995986363608635, −6.24842408461878396296566214830, −5.38861425307692806763428886269, −5.28535667006082211774502065414, −5.24308630160730822218457559836, −5.19670382211094827126225664815, −4.69878047542539799213847256189, −4.64259857386581524762749208680, −4.01013335217070721917736809449, −3.95450526932887332602264172729, −3.92699870702487615962091440827, −3.83994115418674686283762026751, −3.46681132702447006265933140103, −3.06253829825956769926206886081, −2.62845215827761559126014506746, −2.29581768583533008892180805253, −2.25388253093288475349575886833, −2.00809282202571200259223964157, −1.53091810429916511590236429102, −1.24957277443162886590991482531, −0.74638202948797869613309201616, −0.53058713385779782675361403418,
0.53058713385779782675361403418, 0.74638202948797869613309201616, 1.24957277443162886590991482531, 1.53091810429916511590236429102, 2.00809282202571200259223964157, 2.25388253093288475349575886833, 2.29581768583533008892180805253, 2.62845215827761559126014506746, 3.06253829825956769926206886081, 3.46681132702447006265933140103, 3.83994115418674686283762026751, 3.92699870702487615962091440827, 3.95450526932887332602264172729, 4.01013335217070721917736809449, 4.64259857386581524762749208680, 4.69878047542539799213847256189, 5.19670382211094827126225664815, 5.24308630160730822218457559836, 5.28535667006082211774502065414, 5.38861425307692806763428886269, 6.24842408461878396296566214830, 6.26713134262306995986363608635, 6.32502142692066894376224366107, 6.37148812261193787453585518580, 6.70060184641792138131439715967
