| L(s) = 1 | − 2-s + 5-s − 5·7-s + 8-s − 10-s + 13-s + 5·14-s − 16-s − 12·17-s + 10·19-s − 9·23-s − 26-s + 4·31-s + 12·34-s − 5·35-s − 20·37-s − 10·38-s + 40-s − 3·41-s − 8·43-s + 9·46-s − 3·47-s + 7·49-s − 6·53-s − 5·56-s + 9·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.447·5-s − 1.88·7-s + 0.353·8-s − 0.316·10-s + 0.277·13-s + 1.33·14-s − 1/4·16-s − 2.91·17-s + 2.29·19-s − 1.87·23-s − 0.196·26-s + 0.718·31-s + 2.05·34-s − 0.845·35-s − 3.28·37-s − 1.62·38-s + 0.158·40-s − 0.468·41-s − 1.21·43-s + 1.32·46-s − 0.437·47-s + 49-s − 0.824·53-s − 0.668·56-s + 1.17·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2849535931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2849535931\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−10.44508807947744166345766663877, −9.821694936068545818905521958259, −9.725060158320803148184052820647, −9.254145432137583578561163199199, −8.863685700941256608111498434277, −8.477467138535777482818890843878, −8.107057866745498578787661053364, −7.26842081473896458165056238233, −7.05848422069027362511591090621, −6.47648507468338205793467265975, −6.39358349296315908372158223762, −5.74746929713509364362841127180, −5.14699440513647873418509824511, −4.66970194892823826860016908901, −3.98062584586933758135060628074, −3.32257961561926756231082368678, −3.12849365072823452427839587394, −2.11695463703736985782306611517, −1.68876274571149798126048153188, −0.28652670285758625269761238597,
0.28652670285758625269761238597, 1.68876274571149798126048153188, 2.11695463703736985782306611517, 3.12849365072823452427839587394, 3.32257961561926756231082368678, 3.98062584586933758135060628074, 4.66970194892823826860016908901, 5.14699440513647873418509824511, 5.74746929713509364362841127180, 6.39358349296315908372158223762, 6.47648507468338205793467265975, 7.05848422069027362511591090621, 7.26842081473896458165056238233, 8.107057866745498578787661053364, 8.477467138535777482818890843878, 8.863685700941256608111498434277, 9.254145432137583578561163199199, 9.725060158320803148184052820647, 9.821694936068545818905521958259, 10.44508807947744166345766663877
