| L(s) = 1 | − 2·2-s + 2·4-s − 2·7-s + 2·11-s + 12·13-s + 4·14-s − 4·16-s − 34·17-s − 4·22-s + 50·23-s − 25·25-s − 24·26-s − 4·28-s − 34·31-s + 8·32-s + 68·34-s − 36·37-s + 86·41-s − 60·43-s + 4·44-s − 100·46-s + 24·47-s + 2·49-s + 50·50-s + 24·52-s − 98·53-s − 120·61-s + ⋯ |
| L(s) = 1 | − 2-s + 1/2·4-s − 2/7·7-s + 2/11·11-s + 0.923·13-s + 2/7·14-s − 1/4·16-s − 2·17-s − 0.181·22-s + 2.17·23-s − 25-s − 0.923·26-s − 1/7·28-s − 1.09·31-s + 1/4·32-s + 2·34-s − 0.972·37-s + 2.09·41-s − 1.39·43-s + 1/11·44-s − 2.17·46-s + 0.510·47-s + 2/49·49-s + 50-s + 6/13·52-s − 1.84·53-s − 1.96·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4503833900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4503833900\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 50 T + 1250 T^{2} - 50 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1657 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 43 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 60 T + 1800 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 98 T + 4802 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6433 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 60 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 133 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 58 T + 1682 T^{2} + 58 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11906 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 168 T + 14112 T^{2} - 168 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15793 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 216 T + 23328 T^{2} + 216 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.66848591485607308020720816760, −9.444377265547205638621657863118, −9.426726517926128655354009753839, −8.940393555461902552688175178517, −8.897923317453592920422394178747, −8.128606679157993331987775866585, −7.86929119254694419627591068087, −7.19809314042565514020548775496, −6.93342556003032162978763270603, −6.46597435721487574043021379657, −6.07394699908058011245040977475, −5.44176287406336554959928522434, −4.93934767569595533161871144197, −4.15871036518504319332321076941, −4.06176371133925845643854785958, −2.97413328498103161855666255641, −2.82088504011222436300746208985, −1.67166762401514134949426839009, −1.46030807212903527648337970976, −0.27232446414595560541336322330,
0.27232446414595560541336322330, 1.46030807212903527648337970976, 1.67166762401514134949426839009, 2.82088504011222436300746208985, 2.97413328498103161855666255641, 4.06176371133925845643854785958, 4.15871036518504319332321076941, 4.93934767569595533161871144197, 5.44176287406336554959928522434, 6.07394699908058011245040977475, 6.46597435721487574043021379657, 6.93342556003032162978763270603, 7.19809314042565514020548775496, 7.86929119254694419627591068087, 8.128606679157993331987775866585, 8.897923317453592920422394178747, 8.940393555461902552688175178517, 9.426726517926128655354009753839, 9.444377265547205638621657863118, 10.66848591485607308020720816760
