| L(s) = 1 | − 3-s + 9-s + 6·11-s + 3·17-s + 5·19-s + 25-s − 27-s − 6·33-s + 3·41-s + 5·43-s + 2·49-s − 3·51-s − 5·57-s − 12·59-s + 5·67-s − 14·73-s − 75-s + 81-s − 3·89-s − 2·97-s + 6·99-s + 24·107-s + 3·113-s + 14·121-s − 3·123-s + 127-s − 5·129-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.80·11-s + 0.727·17-s + 1.14·19-s + 1/5·25-s − 0.192·27-s − 1.04·33-s + 0.468·41-s + 0.762·43-s + 2/7·49-s − 0.420·51-s − 0.662·57-s − 1.56·59-s + 0.610·67-s − 1.63·73-s − 0.115·75-s + 1/9·81-s − 0.317·89-s − 0.203·97-s + 0.603·99-s + 2.32·107-s + 0.282·113-s + 1.27·121-s − 0.270·123-s + 0.0887·127-s − 0.440·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.352020682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.352020682\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.61619072146803180351107274278, −7.32422567621999186181458748318, −6.85278354150625382127638658286, −6.47309432360768562373488257082, −5.94980494712683988441466278632, −5.75692729802251925801362291311, −5.19925930941201698624733331730, −4.63613235656986934692777973760, −4.26783944619574240034166488210, −3.75420866021664331005836601218, −3.27853030806207598860295079984, −2.77551732002357642454342747808, −1.84126909684414982145330054843, −1.30609692143021771033125671673, −0.73193225387404772207120302717,
0.73193225387404772207120302717, 1.30609692143021771033125671673, 1.84126909684414982145330054843, 2.77551732002357642454342747808, 3.27853030806207598860295079984, 3.75420866021664331005836601218, 4.26783944619574240034166488210, 4.63613235656986934692777973760, 5.19925930941201698624733331730, 5.75692729802251925801362291311, 5.94980494712683988441466278632, 6.47309432360768562373488257082, 6.85278354150625382127638658286, 7.32422567621999186181458748318, 7.61619072146803180351107274278
