| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s + 3·7-s − 4·8-s + 3·9-s + 4·10-s + 8·11-s − 6·12-s − 8·13-s − 6·14-s + 4·15-s + 5·16-s + 17-s − 6·18-s + 9·19-s − 6·20-s − 6·21-s − 16·22-s + 7·23-s + 8·24-s − 2·25-s + 16·26-s − 4·27-s + 9·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 1.13·7-s − 1.41·8-s + 9-s + 1.26·10-s + 2.41·11-s − 1.73·12-s − 2.21·13-s − 1.60·14-s + 1.03·15-s + 5/4·16-s + 0.242·17-s − 1.41·18-s + 2.06·19-s − 1.34·20-s − 1.30·21-s − 3.41·22-s + 1.45·23-s + 1.63·24-s − 2/5·25-s + 3.13·26-s − 0.769·27-s + 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36942084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36942084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223861120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223861120\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 1013 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 15 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9 T + 47 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 79 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 61 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 57 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 15 T + 163 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 21 T + 231 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 137 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 130 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 158 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 213 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−7.948933351871451764592169890768, −7.72661668860088936261327261481, −7.46648728653352703499661086464, −7.38196888872643861009667787474, −7.17116169366642718614309626620, −6.60684164966449480517289377867, −6.16896599446235010377929303476, −5.89879065291237853483047439794, −5.33955855462235612028716243818, −5.14010098620764762232183156530, −4.68981011648269308524935635746, −4.30322864189721234845733213195, −3.70515910155960988106819664274, −3.66271837216641014801264686148, −2.83167031203102196983770404184, −2.38140560357954548243718590884, −1.71704270101475389908264218978, −1.42529511940130741226743444596, −0.76139744220237833636670332256, −0.59478932384227058202086921098,
0.59478932384227058202086921098, 0.76139744220237833636670332256, 1.42529511940130741226743444596, 1.71704270101475389908264218978, 2.38140560357954548243718590884, 2.83167031203102196983770404184, 3.66271837216641014801264686148, 3.70515910155960988106819664274, 4.30322864189721234845733213195, 4.68981011648269308524935635746, 5.14010098620764762232183156530, 5.33955855462235612028716243818, 5.89879065291237853483047439794, 6.16896599446235010377929303476, 6.60684164966449480517289377867, 7.17116169366642718614309626620, 7.38196888872643861009667787474, 7.46648728653352703499661086464, 7.72661668860088936261327261481, 7.948933351871451764592169890768
