| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 5·9-s − 14-s + 16-s + 3·17-s + 5·18-s − 3·23-s − 2·25-s − 28-s − 6·31-s + 32-s + 3·34-s + 5·36-s + 9·41-s − 3·46-s − 15·47-s + 49-s − 2·50-s − 56-s − 6·62-s − 5·63-s + 64-s + 3·68-s + 15·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 5/3·9-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 1.17·18-s − 0.625·23-s − 2/5·25-s − 0.188·28-s − 1.07·31-s + 0.176·32-s + 0.514·34-s + 5/6·36-s + 1.40·41-s − 0.442·46-s − 2.18·47-s + 1/7·49-s − 0.282·50-s − 0.133·56-s − 0.762·62-s − 0.629·63-s + 1/8·64-s + 0.363·68-s + 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.213605989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213605989\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.05678375381477042318874611224, −9.756948225725246098638511465202, −9.502508101509796184785562727000, −8.572471043036661544792716627202, −7.959398509703870639269961294849, −7.40680873284187474787785819703, −7.08975345681600839738665962505, −6.35386202430917741466320291891, −5.93928509818591661972668925621, −5.13656851188741939169424055778, −4.60634381509827538369531600859, −3.84629395034157933487320961776, −3.48844498554738898579609102006, −2.34612635281934047519280732349, −1.42882639006212863066694207280,
1.42882639006212863066694207280, 2.34612635281934047519280732349, 3.48844498554738898579609102006, 3.84629395034157933487320961776, 4.60634381509827538369531600859, 5.13656851188741939169424055778, 5.93928509818591661972668925621, 6.35386202430917741466320291891, 7.08975345681600839738665962505, 7.40680873284187474787785819703, 7.959398509703870639269961294849, 8.572471043036661544792716627202, 9.502508101509796184785562727000, 9.756948225725246098638511465202, 10.05678375381477042318874611224
