| L(s) = 1 | + 4-s − 10·7-s + 12·13-s + 7·25-s − 10·28-s − 18·31-s + 28·37-s + 4·43-s + 61·49-s + 12·52-s − 24·61-s − 64-s − 4·67-s + 20·79-s − 120·91-s − 48·97-s + 7·100-s − 66·103-s + 52·109-s − 13·121-s − 18·124-s + 127-s + 131-s + 137-s + 139-s + 28·148-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.77·7-s + 3.32·13-s + 7/5·25-s − 1.88·28-s − 3.23·31-s + 4.60·37-s + 0.609·43-s + 61/7·49-s + 1.66·52-s − 3.07·61-s − 1/8·64-s − 0.488·67-s + 2.25·79-s − 12.5·91-s − 4.87·97-s + 7/10·100-s − 6.50·103-s + 4.98·109-s − 1.18·121-s − 1.61·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.30·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5825358781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5825358781\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 22 T^{2} - 357 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 65 T^{2} + 2544 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 82 T^{2} + 4515 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 106 T^{2} + 7755 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 134 T^{2} + 11067 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \) |
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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
−6.90821553837668556409642569196, −6.84689493284553215102636646340, −6.32275520573484641073062336203, −6.31034470702351688854249138520, −6.28289165010868604579190601126, −6.11269061315707874675036159956, −5.81930176359591119413123818846, −5.61257972595009024718363304057, −5.42063883476486613082007046270, −5.12393036802517133074895408880, −4.70522864601207437686869297303, −4.14818563812636596751826685681, −4.07595821564026944805670564414, −3.87437324823733129461536360890, −3.76225368890061387537858026232, −3.61974442000099218004413783617, −2.99929666342391830646340757507, −2.95315276108249838189005527675, −2.74166368909118796115629487978, −2.68145849321419947006332481730, −2.12525190572873355167453282042, −1.44466074272957593649031747467, −1.20637852457642661589244141049, −0.988440648607331424708462050876, −0.18305774902831544373234848984,
0.18305774902831544373234848984, 0.988440648607331424708462050876, 1.20637852457642661589244141049, 1.44466074272957593649031747467, 2.12525190572873355167453282042, 2.68145849321419947006332481730, 2.74166368909118796115629487978, 2.95315276108249838189005527675, 2.99929666342391830646340757507, 3.61974442000099218004413783617, 3.76225368890061387537858026232, 3.87437324823733129461536360890, 4.07595821564026944805670564414, 4.14818563812636596751826685681, 4.70522864601207437686869297303, 5.12393036802517133074895408880, 5.42063883476486613082007046270, 5.61257972595009024718363304057, 5.81930176359591119413123818846, 6.11269061315707874675036159956, 6.28289165010868604579190601126, 6.31034470702351688854249138520, 6.32275520573484641073062336203, 6.84689493284553215102636646340, 6.90821553837668556409642569196
