| L(s) = 1 | − 2·3-s + 2·5-s + 3·9-s − 4·11-s − 4·15-s − 25-s − 4·27-s + 8·31-s + 8·33-s − 8·37-s + 6·45-s + 8·47-s − 10·49-s − 8·55-s − 8·59-s + 16·67-s + 16·71-s + 2·75-s + 5·81-s − 12·89-s − 16·93-s − 8·97-s − 12·99-s − 16·103-s + 16·111-s + 5·121-s − 12·125-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 9-s − 1.20·11-s − 1.03·15-s − 1/5·25-s − 0.769·27-s + 1.43·31-s + 1.39·33-s − 1.31·37-s + 0.894·45-s + 1.16·47-s − 1.42·49-s − 1.07·55-s − 1.04·59-s + 1.95·67-s + 1.89·71-s + 0.230·75-s + 5/9·81-s − 1.27·89-s − 1.65·93-s − 0.812·97-s − 1.20·99-s − 1.57·103-s + 1.51·111-s + 5/11·121-s − 1.07·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.69025104683743770288248087931, −6.95500393185725205052266230879, −6.63819944394242819970801103280, −6.41980735036273450707953614266, −5.71315663517628644209063759209, −5.46178499553655492242425903961, −5.21799236645758284504968433033, −4.64216429710905580371490438601, −4.22151360539406005628136168315, −3.52843044714614841013517855455, −2.88975798908926859047599254946, −2.31202893635492901792209170029, −1.75767150129315085093445866929, −0.961636599906691470410823677791, 0,
0.961636599906691470410823677791, 1.75767150129315085093445866929, 2.31202893635492901792209170029, 2.88975798908926859047599254946, 3.52843044714614841013517855455, 4.22151360539406005628136168315, 4.64216429710905580371490438601, 5.21799236645758284504968433033, 5.46178499553655492242425903961, 5.71315663517628644209063759209, 6.41980735036273450707953614266, 6.63819944394242819970801103280, 6.95500393185725205052266230879, 7.69025104683743770288248087931
