| L(s) = 1 | + 3·3-s − 3·7-s + 6·9-s + 13-s − 3·19-s − 9·21-s + 25-s + 9·27-s − 4·31-s + 5·37-s + 3·39-s − 18·43-s − 7·49-s − 9·57-s − 61-s − 18·63-s − 6·67-s − 19·73-s + 3·75-s − 15·79-s + 9·81-s − 3·91-s − 12·93-s + 14·97-s − 15·103-s + 13·109-s + 15·111-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.13·7-s + 2·9-s + 0.277·13-s − 0.688·19-s − 1.96·21-s + 1/5·25-s + 1.73·27-s − 0.718·31-s + 0.821·37-s + 0.480·39-s − 2.74·43-s − 49-s − 1.19·57-s − 0.128·61-s − 2.26·63-s − 0.733·67-s − 2.22·73-s + 0.346·75-s − 1.68·79-s + 81-s − 0.314·91-s − 1.24·93-s + 1.42·97-s − 1.47·103-s + 1.24·109-s + 1.42·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 5 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.86736227589663189605395733420, −7.47508511746324608910238276568, −7.05422603975937911408377381032, −6.58352539774231662932377267189, −6.26073605077007235702901875094, −5.71751556546652999042986469201, −4.99274971054160870798440288555, −4.47301214760946128974044692962, −4.01686432939163183158587700742, −3.42744271327262662638694419453, −3.11958631818888501980600821756, −2.72335668760133611431584720406, −1.91798170586587189613406097007, −1.45302605177861106092667461655, 0,
1.45302605177861106092667461655, 1.91798170586587189613406097007, 2.72335668760133611431584720406, 3.11958631818888501980600821756, 3.42744271327262662638694419453, 4.01686432939163183158587700742, 4.47301214760946128974044692962, 4.99274971054160870798440288555, 5.71751556546652999042986469201, 6.26073605077007235702901875094, 6.58352539774231662932377267189, 7.05422603975937911408377381032, 7.47508511746324608910238276568, 7.86736227589663189605395733420
