| L(s) = 1 | − 3-s − 4-s + 9-s + 12-s − 2·13-s + 16-s − 25-s − 27-s − 36-s + 2·39-s − 8·43-s − 48-s + 2·49-s + 2·52-s + 20·61-s − 64-s + 75-s − 16·79-s + 81-s + 100-s − 8·103-s + 108-s − 2·117-s + 22·121-s + 127-s + 8·129-s + 131-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 1/5·25-s − 0.192·27-s − 1/6·36-s + 0.320·39-s − 1.21·43-s − 0.144·48-s + 2/7·49-s + 0.277·52-s + 2.56·61-s − 1/8·64-s + 0.115·75-s − 1.80·79-s + 1/9·81-s + 1/10·100-s − 0.788·103-s + 0.0962·108-s − 0.184·117-s + 2·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.426961667900432963481636187504, −7.919721723599843316877272792267, −7.34084816164363712403210260669, −7.00261963424523161274508583804, −6.49072089071247518053957852779, −5.99731292871378330820606436826, −5.31474000512286357557971750672, −5.21707378684093067268379875848, −4.50551760684476943665010993257, −4.04867066954241061979433580878, −3.48845598466258731448671350790, −2.74213439026381835441408934743, −2.02117350733386732368742541545, −1.10999263120959718375988906834, 0,
1.10999263120959718375988906834, 2.02117350733386732368742541545, 2.74213439026381835441408934743, 3.48845598466258731448671350790, 4.04867066954241061979433580878, 4.50551760684476943665010993257, 5.21707378684093067268379875848, 5.31474000512286357557971750672, 5.99731292871378330820606436826, 6.49072089071247518053957852779, 7.00261963424523161274508583804, 7.34084816164363712403210260669, 7.919721723599843316877272792267, 8.426961667900432963481636187504
