| L(s) = 1 | − 2·5-s + 4·13-s − 6·17-s − 4·19-s − 4·23-s + 5·25-s − 12·29-s − 8·31-s + 10·37-s − 20·41-s + 24·43-s + 8·47-s + 6·53-s − 4·59-s − 10·61-s − 8·65-s − 12·67-s − 8·71-s + 2·73-s − 8·79-s + 8·83-s + 12·85-s − 6·89-s + 8·95-s − 20·97-s − 10·101-s + 16·107-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.10·13-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 25-s − 2.22·29-s − 1.43·31-s + 1.64·37-s − 3.12·41-s + 3.65·43-s + 1.16·47-s + 0.824·53-s − 0.520·59-s − 1.28·61-s − 0.992·65-s − 1.46·67-s − 0.949·71-s + 0.234·73-s − 0.900·79-s + 0.878·83-s + 1.30·85-s − 0.635·89-s + 0.820·95-s − 2.03·97-s − 0.995·101-s + 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4011738034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4011738034\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 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
−8.684550184011888664313147393708, −8.287473194015390875199421027266, −8.271751151504484456256315239142, −7.36321481715390707455473010977, −7.32114585838750228813597999279, −7.21933097359971682697723695376, −6.48365262117789232112006071306, −6.01150472475383942470278631495, −5.87741474396597399751439054251, −5.54441547913117532361085288489, −4.78518800285213778716035309575, −4.29369951988747574389436800426, −4.26022168705495187138359738218, −3.73910200655359719564867049375, −3.42240605440209132099647609503, −2.76218814417333193748341307604, −2.22615830680149792870735143565, −1.81661694810478825367068465998, −1.15935882014592665747642053826, −0.19865364056486288441435363131,
0.19865364056486288441435363131, 1.15935882014592665747642053826, 1.81661694810478825367068465998, 2.22615830680149792870735143565, 2.76218814417333193748341307604, 3.42240605440209132099647609503, 3.73910200655359719564867049375, 4.26022168705495187138359738218, 4.29369951988747574389436800426, 4.78518800285213778716035309575, 5.54441547913117532361085288489, 5.87741474396597399751439054251, 6.01150472475383942470278631495, 6.48365262117789232112006071306, 7.21933097359971682697723695376, 7.32114585838750228813597999279, 7.36321481715390707455473010977, 8.271751151504484456256315239142, 8.287473194015390875199421027266, 8.684550184011888664313147393708
