| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s − 3·7-s + 8-s + 10-s − 4·11-s + 5·13-s + 3·14-s − 15-s − 16-s − 2·17-s − 19-s − 3·21-s + 4·22-s − 4·23-s + 24-s − 5·26-s − 27-s − 10·29-s + 30-s − 8·31-s − 4·33-s + 2·34-s + 3·35-s − 37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 1.38·13-s + 0.801·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.229·19-s − 0.654·21-s + 0.852·22-s − 0.834·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s − 1.85·29-s + 0.182·30-s − 1.43·31-s − 0.696·33-s + 0.342·34-s + 0.507·35-s − 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 37 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13 T + 86 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.469431670114961762918672593103, −9.320053072555493080117409120304, −8.824643971760549332358762440473, −8.277500539204634583840400575198, −8.117795326298462961192202934998, −7.88481517239625029123951184546, −7.12310953446848890489444829599, −6.89180257601775825436961282401, −6.28913424501088577067100030500, −6.01780284818166202160259939497, −5.22761787903985956667123477545, −5.05068949336486866216640329989, −4.20002343178938403091050157208, −3.58224726861927794961352993671, −3.43510292531433566086146437629, −2.95283374525383741297836899869, −1.80379910684588143234779557876, −1.80206367839233783961982228385, 0, 0,
1.80206367839233783961982228385, 1.80379910684588143234779557876, 2.95283374525383741297836899869, 3.43510292531433566086146437629, 3.58224726861927794961352993671, 4.20002343178938403091050157208, 5.05068949336486866216640329989, 5.22761787903985956667123477545, 6.01780284818166202160259939497, 6.28913424501088577067100030500, 6.89180257601775825436961282401, 7.12310953446848890489444829599, 7.88481517239625029123951184546, 8.117795326298462961192202934998, 8.277500539204634583840400575198, 8.824643971760549332358762440473, 9.320053072555493080117409120304, 9.469431670114961762918672593103
