| L(s) = 1 | − 2·2-s − 3-s + 3·4-s + 2·5-s + 2·6-s + 7-s − 4·8-s + 3·9-s − 4·10-s − 2·11-s − 3·12-s + 4·13-s − 2·14-s − 2·15-s + 5·16-s − 3·17-s − 6·18-s + 7·19-s + 6·20-s − 21-s + 4·22-s − 6·23-s + 4·24-s + 3·25-s − 8·26-s − 8·27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s + 0.377·7-s − 1.41·8-s + 9-s − 1.26·10-s − 0.603·11-s − 0.866·12-s + 1.10·13-s − 0.534·14-s − 0.516·15-s + 5/4·16-s − 0.727·17-s − 1.41·18-s + 1.60·19-s + 1.34·20-s − 0.218·21-s + 0.852·22-s − 1.25·23-s + 0.816·24-s + 3/5·25-s − 1.56·26-s − 1.53·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5906178684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5906178684\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 13 T + 108 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 120 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.65791280221585508799586150832, −13.41226572909749650300357866783, −12.95801060937520669818156462953, −12.18774721888558928121260895031, −11.42377555288379445698982492618, −11.34650080369151156579970477622, −10.72981264238965181959807392919, −9.979494288438638459727049279184, −9.768560529356435508292557762357, −9.390247785900182046123628465206, −8.415837047933552989842722452006, −8.104515689787382320128457509138, −7.35422687001489073777346203906, −6.84241733138923701159262011736, −5.98353161137008289164431147583, −5.72823185862067133238861064161, −4.73978172601008634193291616144, −3.61519141027645554829414607861, −2.28619853611828694232914044186, −1.34474408898394847193415655073,
1.34474408898394847193415655073, 2.28619853611828694232914044186, 3.61519141027645554829414607861, 4.73978172601008634193291616144, 5.72823185862067133238861064161, 5.98353161137008289164431147583, 6.84241733138923701159262011736, 7.35422687001489073777346203906, 8.104515689787382320128457509138, 8.415837047933552989842722452006, 9.390247785900182046123628465206, 9.768560529356435508292557762357, 9.979494288438638459727049279184, 10.72981264238965181959807392919, 11.34650080369151156579970477622, 11.42377555288379445698982492618, 12.18774721888558928121260895031, 12.95801060937520669818156462953, 13.41226572909749650300357866783, 13.65791280221585508799586150832
