| L(s) = 1 | − 2·2-s − 3·3-s + 3·4-s + 3·5-s + 6·6-s − 4·8-s + 6·9-s − 6·10-s + 6·11-s − 9·12-s − 2·13-s − 9·15-s + 5·16-s − 6·17-s − 12·18-s + 7·19-s + 9·20-s − 12·22-s − 3·23-s + 12·24-s + 5·25-s + 4·26-s − 9·27-s − 6·29-s + 18·30-s + 4·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 3/2·4-s + 1.34·5-s + 2.44·6-s − 1.41·8-s + 2·9-s − 1.89·10-s + 1.80·11-s − 2.59·12-s − 0.554·13-s − 2.32·15-s + 5/4·16-s − 1.45·17-s − 2.82·18-s + 1.60·19-s + 2.01·20-s − 2.55·22-s − 0.625·23-s + 2.44·24-s + 25-s + 0.784·26-s − 1.73·27-s − 1.11·29-s + 3.28·30-s + 0.718·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7243196340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7243196340\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.993725791108440706225203381961, −9.964719422467199495669774804698, −9.577938157753465491793756803672, −9.439298907043052881742315132554, −8.710458923345599095197303672315, −8.526481033572757297619183206479, −7.66842074035438143442822113645, −7.21494528859983198412958377385, −6.81265686161049987050336996761, −6.59618667126572169875226055223, −6.03955097018769715981266019783, −5.89063560305509814073844244427, −5.15066986288944884415507512049, −4.89882813185709763161889552555, −4.06917462790875991735629062762, −3.50956076028656827913003797246, −2.46664148168252530134831400305, −1.86938974741709518092253071709, −1.36666969447345380108627382932, −0.62373598491666868618023333025,
0.62373598491666868618023333025, 1.36666969447345380108627382932, 1.86938974741709518092253071709, 2.46664148168252530134831400305, 3.50956076028656827913003797246, 4.06917462790875991735629062762, 4.89882813185709763161889552555, 5.15066986288944884415507512049, 5.89063560305509814073844244427, 6.03955097018769715981266019783, 6.59618667126572169875226055223, 6.81265686161049987050336996761, 7.21494528859983198412958377385, 7.66842074035438143442822113645, 8.526481033572757297619183206479, 8.710458923345599095197303672315, 9.439298907043052881742315132554, 9.577938157753465491793756803672, 9.964719422467199495669774804698, 9.993725791108440706225203381961
