| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 2·7-s − 4·8-s + 3·9-s − 6·12-s + 4·13-s + 4·14-s + 5·16-s + 8·17-s − 6·18-s − 2·19-s + 4·21-s − 4·23-s + 8·24-s − 7·25-s − 8·26-s − 4·27-s − 6·28-s − 4·29-s − 14·31-s − 6·32-s − 16·34-s + 9·36-s − 4·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 1.73·12-s + 1.10·13-s + 1.06·14-s + 5/4·16-s + 1.94·17-s − 1.41·18-s − 0.458·19-s + 0.872·21-s − 0.834·23-s + 1.63·24-s − 7/5·25-s − 1.56·26-s − 0.769·27-s − 1.13·28-s − 0.742·29-s − 2.51·31-s − 1.06·32-s − 2.74·34-s + 3/2·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25826724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25826724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 59 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 92 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 92 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 164 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 180 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 211 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 335 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−7.84676532919604786264199221644, −7.75223637519026616444565387170, −7.36510327369777761432501462440, −7.19567845858334189443649087757, −6.50703573037587965511152699102, −6.25908758173922443441277407605, −5.93770626924424272530581682394, −5.81182530489219458712162590628, −5.24800737694610190785777261723, −5.07897286247835679033243245954, −4.02677784453325349870121378884, −3.95483942437639373668420742458, −3.41214747281056448321994751600, −3.26175567509923831112695400363, −2.21828349113328026131959904455, −2.04434109358615247663584445550, −1.33158997313022529435320164635, −1.04439699014986883960294876400, 0, 0,
1.04439699014986883960294876400, 1.33158997313022529435320164635, 2.04434109358615247663584445550, 2.21828349113328026131959904455, 3.26175567509923831112695400363, 3.41214747281056448321994751600, 3.95483942437639373668420742458, 4.02677784453325349870121378884, 5.07897286247835679033243245954, 5.24800737694610190785777261723, 5.81182530489219458712162590628, 5.93770626924424272530581682394, 6.25908758173922443441277407605, 6.50703573037587965511152699102, 7.19567845858334189443649087757, 7.36510327369777761432501462440, 7.75223637519026616444565387170, 7.84676532919604786264199221644
