| L(s) = 1 | + 2·3-s − 4·7-s + 9-s + 8·13-s + 12·19-s − 8·21-s − 25-s − 4·27-s − 8·31-s + 16·39-s + 12·43-s + 2·49-s + 24·57-s − 4·61-s − 4·63-s + 12·67-s − 4·73-s − 2·75-s + 16·79-s − 11·81-s − 32·91-s − 16·93-s + 4·97-s + 4·103-s + 4·109-s + 8·117-s − 2·121-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s + 1/3·9-s + 2.21·13-s + 2.75·19-s − 1.74·21-s − 1/5·25-s − 0.769·27-s − 1.43·31-s + 2.56·39-s + 1.82·43-s + 2/7·49-s + 3.17·57-s − 0.512·61-s − 0.503·63-s + 1.46·67-s − 0.468·73-s − 0.230·75-s + 1.80·79-s − 1.22·81-s − 3.35·91-s − 1.65·93-s + 0.406·97-s + 0.394·103-s + 0.383·109-s + 0.739·117-s − 0.181·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.994064392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.994064392\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.152149988465093589944033492423, −7.80112274007302981786939825976, −7.30461364522007636941592263794, −7.01745214221863790067433245096, −6.25640894469115166510880960837, −6.06918515683125021534562007845, −5.54249294803330663622247454528, −5.14488371523058796924962906169, −4.15634783241660069999185446339, −3.64860441382983982289597909136, −3.42105723112544346919988444728, −3.11015961583820138455347251131, −2.42770291528322909547625697600, −1.55919757796475639601271333302, −0.802480578218252770751346417786,
0.802480578218252770751346417786, 1.55919757796475639601271333302, 2.42770291528322909547625697600, 3.11015961583820138455347251131, 3.42105723112544346919988444728, 3.64860441382983982289597909136, 4.15634783241660069999185446339, 5.14488371523058796924962906169, 5.54249294803330663622247454528, 6.06918515683125021534562007845, 6.25640894469115166510880960837, 7.01745214221863790067433245096, 7.30461364522007636941592263794, 7.80112274007302981786939825976, 8.152149988465093589944033492423
