| L(s) = 1 | − 2·5-s − 2·7-s + 2·17-s + 2·19-s + 8·23-s − 2·25-s + 10·29-s + 4·31-s + 4·35-s − 4·41-s + 16·43-s − 2·47-s + 3·49-s − 2·53-s − 8·59-s − 16·61-s − 12·67-s + 6·71-s − 24·79-s + 2·83-s − 4·85-s + 4·89-s − 4·95-s + 16·97-s + 14·101-s − 2·107-s + 12·109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 0.485·17-s + 0.458·19-s + 1.66·23-s − 2/5·25-s + 1.85·29-s + 0.718·31-s + 0.676·35-s − 0.624·41-s + 2.43·43-s − 0.291·47-s + 3/7·49-s − 0.274·53-s − 1.04·59-s − 2.04·61-s − 1.46·67-s + 0.712·71-s − 2.70·79-s + 0.219·83-s − 0.433·85-s + 0.423·89-s − 0.410·95-s + 1.62·97-s + 1.39·101-s − 0.193·107-s + 1.14·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.601859547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.601859547\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 146 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 122 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−7.70951484034051205536681935895, −7.52823803322846159759648154169, −7.18889788851027380560146848903, −7.02017564502477463725896315624, −6.34575278849340594344900133085, −6.29853771056009229805266866160, −5.80057983565297474381585157848, −5.63452705621850259837392143914, −4.92740482504015199487720013440, −4.68493701240582109385200006039, −4.41593323584033033673555554056, −4.13135328882348988959716002594, −3.41787713104582311019878868858, −3.17511968326727805533838633187, −3.01659255582841408667145002889, −2.62744838700480254328747733856, −1.91145173289312730176780547763, −1.40009131734888899170382690652, −0.74511465595496916241196092036, −0.52943922930130364121168689504,
0.52943922930130364121168689504, 0.74511465595496916241196092036, 1.40009131734888899170382690652, 1.91145173289312730176780547763, 2.62744838700480254328747733856, 3.01659255582841408667145002889, 3.17511968326727805533838633187, 3.41787713104582311019878868858, 4.13135328882348988959716002594, 4.41593323584033033673555554056, 4.68493701240582109385200006039, 4.92740482504015199487720013440, 5.63452705621850259837392143914, 5.80057983565297474381585157848, 6.29853771056009229805266866160, 6.34575278849340594344900133085, 7.02017564502477463725896315624, 7.18889788851027380560146848903, 7.52823803322846159759648154169, 7.70951484034051205536681935895
