| L(s) = 1 | + 2·3-s + 2·5-s + 7-s + 3·9-s + 7·11-s + 2·13-s + 4·15-s − 17-s + 2·19-s + 2·21-s + 7·23-s + 3·25-s + 4·27-s − 4·29-s + 6·31-s + 14·33-s + 2·35-s − 37-s + 4·39-s − 7·41-s + 4·43-s + 6·45-s − 6·47-s − 9·49-s − 2·51-s + 3·53-s + 14·55-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 − 0.242·17-s + 0.458·19-s + 0.436·21-s + 1.45·23-s + 3/5·25-s + 0.769·27-s − 0.742·29-s + 1.07·31-s + 2.43·33-s + 0.338·35-s − 0.164·37-s + 0.640·39-s − 1.09·41-s + 0.609·43-s + 0.894·45-s − 0.875·47-s − 9/7·49-s − 0.280·51-s + 0.412·53-s + 1.88·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.474953541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.474953541\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 56 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 104 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 106 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 54 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 212 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 144 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−8.815014694413489598571412062105, −8.617290316311767517371589162387, −8.148571135651191144328245380765, −8.033633192173720220709723376374, −7.29846658512509333788672444589, −6.86779272285683299483463009337, −6.63984414580381239027607375686, −6.55608559380942687083707979100, −5.87947871766561781153467074195, −5.34235889094039776567565353095, −5.10141067262799425280761502729, −4.52457942525419756444093845293, −4.06720981588091634387352327185, −3.76657340189447150299955761631, −3.18313748762880687065243988842, −2.99290795806884686627078635784, −2.14692916635221182113908744821, −1.89801378018743872347424133607, −1.20918315853101573553955391569, −0.997917899636925866747111329554,
0.997917899636925866747111329554, 1.20918315853101573553955391569, 1.89801378018743872347424133607, 2.14692916635221182113908744821, 2.99290795806884686627078635784, 3.18313748762880687065243988842, 3.76657340189447150299955761631, 4.06720981588091634387352327185, 4.52457942525419756444093845293, 5.10141067262799425280761502729, 5.34235889094039776567565353095, 5.87947871766561781153467074195, 6.55608559380942687083707979100, 6.63984414580381239027607375686, 6.86779272285683299483463009337, 7.29846658512509333788672444589, 8.033633192173720220709723376374, 8.148571135651191144328245380765, 8.617290316311767517371589162387, 8.815014694413489598571412062105
