| L(s) = 1 | + 2·3-s − 4·7-s + 2·9-s − 12·13-s − 8·21-s + 10·25-s + 6·27-s + 16·31-s − 12·37-s − 24·39-s − 12·43-s + 8·49-s − 24·61-s − 8·63-s − 4·67-s + 4·73-s + 20·75-s + 11·81-s + 48·91-s + 32·93-s + 36·97-s + 4·103-s − 24·111-s − 24·117-s + 4·121-s + 127-s − 24·129-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s + 2/3·9-s − 3.32·13-s − 1.74·21-s + 2·25-s + 1.15·27-s + 2.87·31-s − 1.97·37-s − 3.84·39-s − 1.82·43-s + 8/7·49-s − 3.07·61-s − 1.00·63-s − 0.488·67-s + 0.468·73-s + 2.30·75-s + 11/9·81-s + 5.03·91-s + 3.31·93-s + 3.65·97-s + 0.394·103-s − 2.27·111-s − 2.21·117-s + 4/11·121-s + 0.0887·127-s − 2.11·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6768946904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6768946904\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 238 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 53 | $C_2^3$ | \( 1 + 3598 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.68717685213666173867539920033, −10.53741246784563827321850000183, −10.39772960952919280695046285176, −10.31517972439760862111967515152, −10.30783413356490826296638639362, −9.684992825696092715715463123807, −9.299186274524804035335087601220, −9.122024583398483955409754999430, −9.091176975443855831070908195766, −8.396741378868973744267835407080, −8.141539412500296434633534646016, −7.84264132045831719729779735590, −7.41727364560654563618102525190, −6.92382608257770560132859887593, −6.86445969136353655626133935062, −6.54806518347674944274692276756, −6.14693231226485150864417260201, −5.33541712393311827841809943575, −5.00569969556774291250750573922, −4.54704649572272101623991758334, −4.50603179780288941406115005502, −3.21855912675630881585355836082, −3.18642659840942046056459666008, −2.80493384870982070917755980699, −2.15777231959226116393597609132,
2.15777231959226116393597609132, 2.80493384870982070917755980699, 3.18642659840942046056459666008, 3.21855912675630881585355836082, 4.50603179780288941406115005502, 4.54704649572272101623991758334, 5.00569969556774291250750573922, 5.33541712393311827841809943575, 6.14693231226485150864417260201, 6.54806518347674944274692276756, 6.86445969136353655626133935062, 6.92382608257770560132859887593, 7.41727364560654563618102525190, 7.84264132045831719729779735590, 8.141539412500296434633534646016, 8.396741378868973744267835407080, 9.091176975443855831070908195766, 9.122024583398483955409754999430, 9.299186274524804035335087601220, 9.684992825696092715715463123807, 10.30783413356490826296638639362, 10.31517972439760862111967515152, 10.39772960952919280695046285176, 10.53741246784563827321850000183, 11.68717685213666173867539920033
