| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s + 9-s − 10-s + 6·13-s + 16-s + 10·17-s + 18-s − 20-s − 4·25-s + 6·26-s − 5·29-s + 32-s + 10·34-s + 36-s + 11·37-s − 40-s + 10·41-s − 45-s − 5·49-s − 4·50-s + 6·52-s − 10·53-s − 5·58-s + 11·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.66·13-s + 1/4·16-s + 2.42·17-s + 0.235·18-s − 0.223·20-s − 4/5·25-s + 1.17·26-s − 0.928·29-s + 0.176·32-s + 1.71·34-s + 1/6·36-s + 1.80·37-s − 0.158·40-s + 1.56·41-s − 0.149·45-s − 5/7·49-s − 0.565·50-s + 0.832·52-s − 1.37·53-s − 0.656·58-s + 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.104964817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.104964817\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.81530839800383487204844214728, −7.58597684354092766829123996559, −7.36941811646773336654414406383, −6.50892715772454504342662700395, −6.07830237495320569941315571411, −5.91934491513942361032553767019, −5.44780214858041481982822081593, −4.83107423436579488071040934704, −4.32407261734453635202004720678, −3.72563320181881104358908276390, −3.56942757613183001924271165240, −3.03521698397683129461770642663, −2.25406782185316202233815114242, −1.42080191061395483829455265875, −0.912061386349406761181507385199,
0.912061386349406761181507385199, 1.42080191061395483829455265875, 2.25406782185316202233815114242, 3.03521698397683129461770642663, 3.56942757613183001924271165240, 3.72563320181881104358908276390, 4.32407261734453635202004720678, 4.83107423436579488071040934704, 5.44780214858041481982822081593, 5.91934491513942361032553767019, 6.07830237495320569941315571411, 6.50892715772454504342662700395, 7.36941811646773336654414406383, 7.58597684354092766829123996559, 7.81530839800383487204844214728
