L(s) = 1 | + 2-s + 4-s − 5·5-s + 8-s + 9-s − 5·10-s − 5·11-s + 16-s − 3·17-s + 18-s − 5·20-s − 5·22-s + 9·25-s − 5·29-s + 32-s − 3·34-s + 36-s + 5·37-s − 5·40-s − 5·44-s − 5·45-s + 47-s − 5·49-s + 9·50-s + 25·55-s − 5·58-s + 10·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 2.23·5-s + 0.353·8-s + 1/3·9-s − 1.58·10-s − 1.50·11-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 1.11·20-s − 1.06·22-s + 9/5·25-s − 0.928·29-s + 0.176·32-s − 0.514·34-s + 1/6·36-s + 0.821·37-s − 0.790·40-s − 0.753·44-s − 0.745·45-s + 0.145·47-s − 5/7·49-s + 1.27·50-s + 3.37·55-s − 0.656·58-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9974555954\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9974555954\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 120 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
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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
−8.448808625167248317519136153704, −8.247407298834868330033554347961, −7.69335932532739626710245665397, −7.36658521776595662629876448679, −7.19353894966924005518263491801, −6.40652225141586337841427985932, −5.86574330715237103383327635881, −5.26576718936508145202374698159, −4.63806379063330557913180776046, −4.43227128971447294513811917118, −3.72231408248868165895806106006, −3.44832995233677547602503487011, −2.68381781904019296952631234986, −1.98881999771216838877484296215, −0.49060658600177524445168074707,
0.49060658600177524445168074707, 1.98881999771216838877484296215, 2.68381781904019296952631234986, 3.44832995233677547602503487011, 3.72231408248868165895806106006, 4.43227128971447294513811917118, 4.63806379063330557913180776046, 5.26576718936508145202374698159, 5.86574330715237103383327635881, 6.40652225141586337841427985932, 7.19353894966924005518263491801, 7.36658521776595662629876448679, 7.69335932532739626710245665397, 8.247407298834868330033554347961, 8.448808625167248317519136153704