| L(s) = 1 | + 2-s + 4-s + 5·7-s + 8-s − 9-s + 5·14-s + 16-s + 4·17-s − 18-s + 7·23-s + 25-s + 5·28-s − 6·31-s + 32-s + 4·34-s − 36-s − 7·41-s + 7·46-s − 10·47-s + 18·49-s + 50-s + 5·56-s − 6·62-s − 5·63-s + 64-s + 4·68-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.88·7-s + 0.353·8-s − 1/3·9-s + 1.33·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.45·23-s + 1/5·25-s + 0.944·28-s − 1.07·31-s + 0.176·32-s + 0.685·34-s − 1/6·36-s − 1.09·41-s + 1.03·46-s − 1.45·47-s + 18/7·49-s + 0.141·50-s + 0.668·56-s − 0.762·62-s − 0.629·63-s + 1/8·64-s + 0.485·68-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.295749689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.295749689\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−9.071632957971613101936898809269, −8.647480107767751505389909368384, −8.344153947400599392781736492365, −7.68465119716288296707030029089, −7.33995934573308063189124840559, −6.93734611049548710617530575469, −6.10262830914848717945300813256, −5.60852842297404444454152605292, −5.07738198069100610634530407319, −4.84055207956246431338218165898, −4.18027081488451575062372386671, −3.40140486293295855807991098815, −2.86317249015929539126109241313, −1.87951807381035946762005813652, −1.28729413496941365951113077853,
1.28729413496941365951113077853, 1.87951807381035946762005813652, 2.86317249015929539126109241313, 3.40140486293295855807991098815, 4.18027081488451575062372386671, 4.84055207956246431338218165898, 5.07738198069100610634530407319, 5.60852842297404444454152605292, 6.10262830914848717945300813256, 6.93734611049548710617530575469, 7.33995934573308063189124840559, 7.68465119716288296707030029089, 8.344153947400599392781736492365, 8.647480107767751505389909368384, 9.071632957971613101936898809269
