| L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s − 3·8-s − 9-s + 4·10-s + 2·11-s − 12-s + 9·13-s + 4·15-s + 16-s − 6·17-s + 18-s − 6·19-s − 4·20-s − 2·22-s − 11·23-s + 3·24-s + 2·25-s − 9·26-s + 7·29-s − 4·30-s − 31-s + 32-s − 2·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 1.06·8-s − 1/3·9-s + 1.26·10-s + 0.603·11-s − 0.288·12-s + 2.49·13-s + 1.03·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 1.37·19-s − 0.894·20-s − 0.426·22-s − 2.29·23-s + 0.612·24-s + 2/5·25-s − 1.76·26-s + 1.29·29-s − 0.730·30-s − 0.179·31-s + 0.176·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6568969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6568969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 233 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 9 T + 42 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 11 T + 72 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 50 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 47 | $C_4$ | \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 92 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 15 T + 174 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 23 T + 262 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - T + 108 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 99 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 111 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.450604791572393679103706088015, −8.402549637107574956424854878358, −8.015995747735520059471446669805, −7.971769487660667920387505178765, −6.99544047811569038751910363621, −6.78939044524651179190248989962, −6.35293729329555191142614821785, −6.15626163530305620864104920611, −5.94991921897196257664520266983, −5.20381837538238581853471767685, −4.53836472590573982638283282966, −4.21584544103533525191343950012, −3.74528814558688739272155065076, −3.64954341633733369771329806398, −3.07038186759773682238685262217, −2.03897691129650339881478455056, −1.98681198031993093905691569293, −0.934305716325276894080474290353, 0, 0,
0.934305716325276894080474290353, 1.98681198031993093905691569293, 2.03897691129650339881478455056, 3.07038186759773682238685262217, 3.64954341633733369771329806398, 3.74528814558688739272155065076, 4.21584544103533525191343950012, 4.53836472590573982638283282966, 5.20381837538238581853471767685, 5.94991921897196257664520266983, 6.15626163530305620864104920611, 6.35293729329555191142614821785, 6.78939044524651179190248989962, 6.99544047811569038751910363621, 7.971769487660667920387505178765, 8.015995747735520059471446669805, 8.402549637107574956424854878358, 8.450604791572393679103706088015
