| L(s) = 1 | + 2·2-s + 3·4-s − 3·5-s + 5·7-s + 4·8-s − 6·10-s + 3·11-s − 2·13-s + 10·14-s + 5·16-s − 6·17-s − 2·19-s − 9·20-s + 6·22-s + 6·23-s + 5·25-s − 4·26-s + 15·28-s − 9·29-s − 14·31-s + 6·32-s − 12·34-s − 15·35-s + 10·37-s − 4·38-s − 12·40-s + 4·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.34·5-s + 1.88·7-s + 1.41·8-s − 1.89·10-s + 0.904·11-s − 0.554·13-s + 2.67·14-s + 5/4·16-s − 1.45·17-s − 0.458·19-s − 2.01·20-s + 1.27·22-s + 1.25·23-s + 25-s − 0.784·26-s + 2.83·28-s − 1.67·29-s − 2.51·31-s + 1.06·32-s − 2.05·34-s − 2.53·35-s + 1.64·37-s − 0.648·38-s − 1.89·40-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.312471756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.312471756\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−10.48540264147949102059502109763, −9.283595896992974442075476352530, −9.093063828339997281485906691852, −8.929852928436489647698857220375, −8.225506060863189531468402953674, −7.70767436708935419406150857337, −7.38724254031940482023510481459, −7.26367851897535541245973555076, −6.82444816796319739797445701865, −5.98214044708989093621057103376, −5.74691084702890531355335921589, −5.15859272159296445261864867054, −4.69444558017019251894814330617, −4.36619023701247332718681478685, −3.95060500408206712452364208494, −3.76474517141537055112774050366, −2.86528573219665802794138618280, −2.16680933705359737953008220227, −1.84675008184175019533426816968, −0.818474004284224395541938613026,
0.818474004284224395541938613026, 1.84675008184175019533426816968, 2.16680933705359737953008220227, 2.86528573219665802794138618280, 3.76474517141537055112774050366, 3.95060500408206712452364208494, 4.36619023701247332718681478685, 4.69444558017019251894814330617, 5.15859272159296445261864867054, 5.74691084702890531355335921589, 5.98214044708989093621057103376, 6.82444816796319739797445701865, 7.26367851897535541245973555076, 7.38724254031940482023510481459, 7.70767436708935419406150857337, 8.225506060863189531468402953674, 8.929852928436489647698857220375, 9.093063828339997281485906691852, 9.283595896992974442075476352530, 10.48540264147949102059502109763
