L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 9-s − 6·11-s + 4·12-s − 4·16-s + 8·17-s + 2·18-s − 19-s − 12·22-s − 4·25-s − 4·27-s − 8·32-s − 12·33-s + 16·34-s + 2·36-s − 2·38-s + 6·41-s − 5·43-s − 12·44-s − 8·48-s + 12·49-s − 8·50-s + 16·51-s − 8·54-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 1/3·9-s − 1.80·11-s + 1.15·12-s − 16-s + 1.94·17-s + 0.471·18-s − 0.229·19-s − 2.55·22-s − 4/5·25-s − 0.769·27-s − 1.41·32-s − 2.08·33-s + 2.74·34-s + 1/3·36-s − 0.324·38-s + 0.937·41-s − 0.762·43-s − 1.80·44-s − 1.15·48-s + 12/7·49-s − 1.13·50-s + 2.24·51-s − 1.08·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.827696588\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.827696588\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−11.00990148479361974512903595386, −10.27410735899096128663491795846, −10.00043929837586322169820950191, −9.273942511595754407638758356055, −8.603798975047170577630500142518, −8.113977769356517947593134361678, −7.52916963272873128285349722612, −7.17099812602487980015996071865, −5.88534826524461339567484206573, −5.72252595021001812577450085372, −5.04939334556222462380948801741, −4.19774560068835641601909889463, −3.46509995586147942027305824486, −2.89889286302378118352656738774, −2.23251610210627076672437444264,
2.23251610210627076672437444264, 2.89889286302378118352656738774, 3.46509995586147942027305824486, 4.19774560068835641601909889463, 5.04939334556222462380948801741, 5.72252595021001812577450085372, 5.88534826524461339567484206573, 7.17099812602487980015996071865, 7.52916963272873128285349722612, 8.113977769356517947593134361678, 8.603798975047170577630500142518, 9.273942511595754407638758356055, 10.00043929837586322169820950191, 10.27410735899096128663491795846, 11.00990148479361974512903595386