| L(s) = 1 | − 3·5-s + 7-s − 3·11-s − 6·17-s + 2·19-s + 25-s − 18·29-s − 5·31-s − 3·35-s − 10·37-s + 12·47-s − 6·49-s − 9·53-s + 9·55-s − 9·59-s − 24·67-s − 12·73-s − 3·77-s + 9·79-s − 6·83-s + 18·85-s + 6·89-s − 6·95-s + 24·101-s + 4·103-s + 21·107-s + 4·109-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s − 0.904·11-s − 1.45·17-s + 0.458·19-s + 1/5·25-s − 3.34·29-s − 0.898·31-s − 0.507·35-s − 1.64·37-s + 1.75·47-s − 6/7·49-s − 1.23·53-s + 1.21·55-s − 1.17·59-s − 2.93·67-s − 1.40·73-s − 0.341·77-s + 1.01·79-s − 0.658·83-s + 1.95·85-s + 0.635·89-s − 0.615·95-s + 2.38·101-s + 0.394·103-s + 2.03·107-s + 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1924688187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1924688187\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 24 T + 259 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−10.63003244423538236457362486274, −9.648379453631351592936606508207, −9.196584443765097739552033397841, −8.912671000369812131225293491379, −8.623437817106801739532009345320, −7.85402883432371423855138908813, −7.67899035872686296459908945376, −7.25115737345175603387913699904, −7.23406371073138902944668589510, −6.24437818555985477889632961342, −5.92891124247341074304070858059, −5.31356309469058832063843697828, −4.97978908789291526217131863315, −4.29048662291944061885694971610, −4.07643661437941303177221755672, −3.35970825456074349191837411546, −3.08870515957561331948077713993, −1.97355284828027452590557258020, −1.79801334137394120496675071308, −0.19068555869781792360466605670,
0.19068555869781792360466605670, 1.79801334137394120496675071308, 1.97355284828027452590557258020, 3.08870515957561331948077713993, 3.35970825456074349191837411546, 4.07643661437941303177221755672, 4.29048662291944061885694971610, 4.97978908789291526217131863315, 5.31356309469058832063843697828, 5.92891124247341074304070858059, 6.24437818555985477889632961342, 7.23406371073138902944668589510, 7.25115737345175603387913699904, 7.67899035872686296459908945376, 7.85402883432371423855138908813, 8.623437817106801739532009345320, 8.912671000369812131225293491379, 9.196584443765097739552033397841, 9.648379453631351592936606508207, 10.63003244423538236457362486274
