L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 16-s + 2·17-s + 2·20-s + 3·25-s − 6·29-s − 32-s − 2·34-s − 6·37-s − 2·40-s − 8·41-s + 2·49-s − 3·50-s + 24·53-s + 6·58-s + 14·61-s + 64-s + 2·68-s − 8·73-s + 6·74-s + 2·80-s − 9·81-s + 8·82-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 1/4·16-s + 0.485·17-s + 0.447·20-s + 3/5·25-s − 1.11·29-s − 0.176·32-s − 0.342·34-s − 0.986·37-s − 0.316·40-s − 1.24·41-s + 2/7·49-s − 0.424·50-s + 3.29·53-s + 0.787·58-s + 1.79·61-s + 1/8·64-s + 0.242·68-s − 0.936·73-s + 0.697·74-s + 0.223·80-s − 81-s + 0.883·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.670848242\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.670848242\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.230943876419836394281235403492, −7.55469839025532965574521483350, −7.09805028996230366611800202439, −7.01610200686037364296802782414, −6.35784837713742203074995971095, −5.76849242620884454938523835340, −5.57573398756874914439708837487, −5.12540704560261718618391983344, −4.46500723080437507014814351574, −3.72362229075729027762429082542, −3.40212866449501956160930369280, −2.58727740316200920401769919761, −2.10733963861136565367472916595, −1.54603992608396505332235677158, −0.66828474417584780230645184416,
0.66828474417584780230645184416, 1.54603992608396505332235677158, 2.10733963861136565367472916595, 2.58727740316200920401769919761, 3.40212866449501956160930369280, 3.72362229075729027762429082542, 4.46500723080437507014814351574, 5.12540704560261718618391983344, 5.57573398756874914439708837487, 5.76849242620884454938523835340, 6.35784837713742203074995971095, 7.01610200686037364296802782414, 7.09805028996230366611800202439, 7.55469839025532965574521483350, 8.230943876419836394281235403492