| L(s) = 1 | + 2·2-s + 5·5-s − 14·7-s − 8·8-s + 10·10-s − 6·11-s − 68·13-s − 28·14-s − 16·16-s + 156·17-s + 88·19-s − 12·22-s − 120·23-s − 136·26-s − 126·29-s + 244·31-s + 312·34-s − 70·35-s − 608·37-s + 176·38-s − 40·40-s + 480·41-s − 104·43-s − 240·46-s − 600·47-s + 343·49-s − 516·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s − 0.164·11-s − 1.45·13-s − 0.534·14-s − 1/4·16-s + 2.22·17-s + 1.06·19-s − 0.116·22-s − 1.08·23-s − 1.02·26-s − 0.806·29-s + 1.41·31-s + 1.57·34-s − 0.338·35-s − 2.70·37-s + 0.751·38-s − 0.158·40-s + 1.82·41-s − 0.368·43-s − 0.769·46-s − 1.86·47-s + 49-s − 1.33·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.415106883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.415106883\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 p T - 3 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T - 1295 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 68 T + 2427 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 78 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 44 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 120 T + 2233 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 126 T - 8513 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 244 T + 29745 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 304 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 480 T + 161479 T^{2} - 480 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 104 T - 68691 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 600 T + 256177 T^{2} + 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 258 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 534 T + 79777 T^{2} + 534 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 362 T - 95937 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 p T - 51 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 972 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 470 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1244 T + 1054497 T^{2} + 1244 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 396 T - 414971 T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 972 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 46 T - 910557 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.07515962067977572533162886955, −9.654815656473822269402845699659, −9.545811076780990486209945250317, −8.821109965430760364129334379176, −8.401728481077574997744826076223, −7.66864431526888342118215882008, −7.51340396021370086516600660038, −7.18414038448278547527218492124, −6.34557156976400250152855756372, −6.10101231172062599908926530671, −5.53635420023510681536322881769, −5.30195790978287636700409965502, −4.72993058818170728910832964292, −4.26999220691637587166254516810, −3.42572469535070706395919132657, −3.17968935316439485258884916323, −2.79179759940288663010719156394, −1.88304406892896568762972992588, −1.30458702128427782373890446337, −0.28124779898681569155790539712,
0.28124779898681569155790539712, 1.30458702128427782373890446337, 1.88304406892896568762972992588, 2.79179759940288663010719156394, 3.17968935316439485258884916323, 3.42572469535070706395919132657, 4.26999220691637587166254516810, 4.72993058818170728910832964292, 5.30195790978287636700409965502, 5.53635420023510681536322881769, 6.10101231172062599908926530671, 6.34557156976400250152855756372, 7.18414038448278547527218492124, 7.51340396021370086516600660038, 7.66864431526888342118215882008, 8.401728481077574997744826076223, 8.821109965430760364129334379176, 9.545811076780990486209945250317, 9.654815656473822269402845699659, 10.07515962067977572533162886955
