| L(s) = 1 | − 2·3-s + 4-s + 5-s − 5·7-s + 4·9-s + 11-s − 2·12-s − 5·13-s − 2·15-s + 16-s − 3·17-s − 19-s + 20-s + 10·21-s + 8·23-s − 5·27-s − 5·28-s + 3·29-s − 4·31-s − 2·33-s − 5·35-s + 4·36-s + 6·37-s + 10·39-s + 41-s + 44-s + 4·45-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.447·5-s − 1.88·7-s + 4/3·9-s + 0.301·11-s − 0.577·12-s − 1.38·13-s − 0.516·15-s + 1/4·16-s − 0.727·17-s − 0.229·19-s + 0.223·20-s + 2.18·21-s + 1.66·23-s − 0.962·27-s − 0.944·28-s + 0.557·29-s − 0.718·31-s − 0.348·33-s − 0.845·35-s + 2/3·36-s + 0.986·37-s + 1.60·39-s + 0.156·41-s + 0.150·44-s + 0.596·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1164 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1164 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4257850601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4257850601\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 7 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T - 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T - 25 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - T - 44 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 13 T + 107 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - T - 27 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−19.5343565411, −19.2719168776, −18.7288871400, −17.9913054374, −17.2916467970, −16.9847376956, −16.4378401131, −16.0152667056, −15.3407624917, −14.8808306379, −13.8875429389, −13.0274105677, −12.7456410012, −12.2881325430, −11.4681948969, −10.7625112973, −10.1669690688, −9.55599430320, −9.04597018324, −7.40608181411, −6.85113387957, −6.35959394243, −5.50825423021, −4.40214362049, −2.83188041549,
2.83188041549, 4.40214362049, 5.50825423021, 6.35959394243, 6.85113387957, 7.40608181411, 9.04597018324, 9.55599430320, 10.1669690688, 10.7625112973, 11.4681948969, 12.2881325430, 12.7456410012, 13.0274105677, 13.8875429389, 14.8808306379, 15.3407624917, 16.0152667056, 16.4378401131, 16.9847376956, 17.2916467970, 17.9913054374, 18.7288871400, 19.2719168776, 19.5343565411
