| L(s) = 1 | − 4-s − 4·5-s + 7-s − 2·13-s − 3·16-s + 4·20-s − 2·23-s + 2·25-s − 28-s − 4·29-s − 4·35-s − 4·49-s + 2·52-s − 12·53-s − 2·59-s + 7·64-s + 8·65-s − 14·67-s + 12·80-s + 12·83-s − 2·91-s + 2·92-s − 2·100-s + 22·103-s + 4·107-s − 3·112-s + 8·115-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.78·5-s + 0.377·7-s − 0.554·13-s − 3/4·16-s + 0.894·20-s − 0.417·23-s + 2/5·25-s − 0.188·28-s − 0.742·29-s − 0.676·35-s − 4/7·49-s + 0.277·52-s − 1.64·53-s − 0.260·59-s + 7/8·64-s + 0.992·65-s − 1.71·67-s + 1.34·80-s + 1.31·83-s − 0.209·91-s + 0.208·92-s − 1/5·100-s + 2.16·103-s + 0.386·107-s − 0.283·112-s + 0.746·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476847 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476847 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3678986828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3678986828\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
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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
−8.518012193326083168239773684053, −7.922583135871710699278156799943, −7.73021188260856429978186167244, −7.40904643954749710320830390130, −6.86338967489922307703618982039, −6.22952845655889735033383795376, −5.78128467339301927976011904932, −4.93069345780551210520769825805, −4.71702408908334060822914263810, −4.24226373072605315200529813820, −3.68951465212597269004105272195, −3.32010483694450075046205667643, −2.42438281002969834578160968540, −1.65650976503743873445129429590, −0.32061397262771802613433824103,
0.32061397262771802613433824103, 1.65650976503743873445129429590, 2.42438281002969834578160968540, 3.32010483694450075046205667643, 3.68951465212597269004105272195, 4.24226373072605315200529813820, 4.71702408908334060822914263810, 4.93069345780551210520769825805, 5.78128467339301927976011904932, 6.22952845655889735033383795376, 6.86338967489922307703618982039, 7.40904643954749710320830390130, 7.73021188260856429978186167244, 7.922583135871710699278156799943, 8.518012193326083168239773684053
