| L(s) = 1 | + 2·3-s + 4-s + 6·5-s + 3·9-s + 5·11-s + 2·12-s + 12·15-s + 16-s + 6·20-s + 8·23-s + 17·25-s + 4·27-s − 2·31-s + 10·33-s + 3·36-s − 12·37-s + 5·44-s + 18·45-s − 2·47-s + 2·48-s − 10·49-s + 12·53-s + 30·55-s − 20·59-s + 12·60-s + 64-s − 6·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 2.68·5-s + 9-s + 1.50·11-s + 0.577·12-s + 3.09·15-s + 1/4·16-s + 1.34·20-s + 1.66·23-s + 17/5·25-s + 0.769·27-s − 0.359·31-s + 1.74·33-s + 1/2·36-s − 1.97·37-s + 0.753·44-s + 2.68·45-s − 0.291·47-s + 0.288·48-s − 1.42·49-s + 1.64·53-s + 4.04·55-s − 2.60·59-s + 1.54·60-s + 1/8·64-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4186116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4186116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.01198512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.01198512\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
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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
−7.36260827061863180158325670551, −6.80928325637438186879603867982, −6.62475750166554661041789766388, −6.25251476412103645777598539699, −5.89423980338579260379198124792, −5.25566528367566277200195898472, −5.13289058297610594749512926364, −4.46084992496455703078688833791, −3.87149777111293165154979015020, −3.21935054931024025505407094893, −3.03107665456399427224887516959, −2.38260029833515102488323978819, −1.82338902456101278198546006187, −1.62830424201290219784693302685, −1.15376663276338816928658601222,
1.15376663276338816928658601222, 1.62830424201290219784693302685, 1.82338902456101278198546006187, 2.38260029833515102488323978819, 3.03107665456399427224887516959, 3.21935054931024025505407094893, 3.87149777111293165154979015020, 4.46084992496455703078688833791, 5.13289058297610594749512926364, 5.25566528367566277200195898472, 5.89423980338579260379198124792, 6.25251476412103645777598539699, 6.62475750166554661041789766388, 6.80928325637438186879603867982, 7.36260827061863180158325670551
