| L(s) = 1 | + 3-s + 3·7-s − 2·9-s + 5·13-s + 7·19-s + 3·21-s − 7·25-s − 5·27-s + 4·31-s + 9·37-s + 5·39-s + 2·43-s − 7·49-s + 7·57-s + 11·61-s − 6·63-s − 10·67-s + 13·73-s − 7·75-s − 9·79-s + 81-s + 15·91-s + 4·93-s − 2·97-s − 25·103-s + 9·109-s + 9·111-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s − 2/3·9-s + 1.38·13-s + 1.60·19-s + 0.654·21-s − 7/5·25-s − 0.962·27-s + 0.718·31-s + 1.47·37-s + 0.800·39-s + 0.304·43-s − 49-s + 0.927·57-s + 1.40·61-s − 0.755·63-s − 1.22·67-s + 1.52·73-s − 0.808·75-s − 1.01·79-s + 1/9·81-s + 1.57·91-s + 0.414·93-s − 0.203·97-s − 2.46·103-s + 0.862·109-s + 0.854·111-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
| \(L(1)\) |
\(\approx\) |
\(3.245342418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.245342418\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.071961672527906863140337165001, −7.88397742060837631683710051396, −7.27708068995388789561234657196, −6.84968875481701167403000759646, −6.11122279755619925536658942099, −5.81311481108806217063450475162, −5.52850166800920951682541086700, −4.84574940770026474475340192315, −4.40660693366901712230766505898, −3.84015841320905354650239051908, −3.33489208676285362956347948895, −2.89020100760438059253021076952, −2.15312807742855363333262914237, −1.56284056022107850365386695135, −0.836757933859795967155902860058,
0.836757933859795967155902860058, 1.56284056022107850365386695135, 2.15312807742855363333262914237, 2.89020100760438059253021076952, 3.33489208676285362956347948895, 3.84015841320905354650239051908, 4.40660693366901712230766505898, 4.84574940770026474475340192315, 5.52850166800920951682541086700, 5.81311481108806217063450475162, 6.11122279755619925536658942099, 6.84968875481701167403000759646, 7.27708068995388789561234657196, 7.88397742060837631683710051396, 8.071961672527906863140337165001
