| L(s) = 1 | + 2·3-s − 2·5-s − 7-s + 3·9-s − 7·11-s − 2·13-s − 4·15-s − 5·17-s − 2·19-s − 2·21-s + 9·23-s + 3·25-s + 4·27-s + 6·29-s − 14·33-s + 2·35-s − 13·37-s − 4·39-s + 41-s + 10·43-s − 6·45-s + 8·47-s − 5·49-s − 10·51-s + 3·53-s + 14·55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.377·7-s + 9-s − 2.11·11-s − 0.554·13-s − 1.03·15-s − 1.21·17-s − 0.458·19-s − 0.436·21-s + 1.87·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s − 2.43·33-s + 0.338·35-s − 2.13·37-s − 0.640·39-s + 0.156·41-s + 1.52·43-s − 0.894·45-s + 1.16·47-s − 5/7·49-s − 1.40·51-s + 0.412·53-s + 1.88·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.850022275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.850022275\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 13 T + 108 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 144 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 17 T + 206 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + T + 150 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 13 T + 212 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T + 192 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−8.187110035705169101670671685782, −8.030400266182332960845686872523, −7.51058930695153136807886394494, −7.16470395155086891787398953434, −6.98484170726586326853759547747, −6.75033611688615411215862387535, −6.19708465351487138075673695168, −5.62407318754933667129268468914, −5.19176439609668699958773523392, −4.97193819103969118621659914529, −4.48575379805346911251739604153, −4.37596372168201113290450963265, −3.52733146960562458211110208164, −3.50556840931358384532802721418, −2.84136018688262213149539336513, −2.73523461715244166097020312309, −2.08090317349652672795486402089, −2.02396729287348212223346775958, −0.74073615288416522416703799440, −0.54768202888856725153124558672,
0.54768202888856725153124558672, 0.74073615288416522416703799440, 2.02396729287348212223346775958, 2.08090317349652672795486402089, 2.73523461715244166097020312309, 2.84136018688262213149539336513, 3.50556840931358384532802721418, 3.52733146960562458211110208164, 4.37596372168201113290450963265, 4.48575379805346911251739604153, 4.97193819103969118621659914529, 5.19176439609668699958773523392, 5.62407318754933667129268468914, 6.19708465351487138075673695168, 6.75033611688615411215862387535, 6.98484170726586326853759547747, 7.16470395155086891787398953434, 7.51058930695153136807886394494, 8.030400266182332960845686872523, 8.187110035705169101670671685782
