L(s) = 1 | − 3·3-s + 3·7-s + 6·9-s − 6·13-s + 19-s − 9·21-s − 25-s − 9·27-s + 7·31-s − 7·37-s + 18·39-s + 2·49-s − 3·57-s − 2·61-s + 18·63-s − 67-s − 7·73-s + 3·75-s − 7·79-s + 9·81-s − 18·91-s − 21·93-s + 12·97-s + 17·103-s − 7·109-s + 21·111-s − 36·117-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 1.13·7-s + 2·9-s − 1.66·13-s + 0.229·19-s − 1.96·21-s − 1/5·25-s − 1.73·27-s + 1.25·31-s − 1.15·37-s + 2.88·39-s + 2/7·49-s − 0.397·57-s − 0.256·61-s + 2.26·63-s − 0.122·67-s − 0.819·73-s + 0.346·75-s − 0.787·79-s + 81-s − 1.88·91-s − 2.17·93-s + 1.21·97-s + 1.67·103-s − 0.670·109-s + 1.99·111-s − 3.32·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 77 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.277874377956488697303012697109, −7.71410731117866147437499698865, −7.35450248143790297800467637114, −6.99607528598158030816254869546, −6.43932913120193562062280154793, −5.97548267298282443053485818395, −5.38863418128718444736954436328, −5.09638095647658878650608673722, −4.58526120228103052961464181371, −4.41585586689347084490462347489, −3.48408447439410775348355100635, −2.59838524630422362465087088833, −1.87278980973687933067061252334, −1.10502185812500183932077865075, 0,
1.10502185812500183932077865075, 1.87278980973687933067061252334, 2.59838524630422362465087088833, 3.48408447439410775348355100635, 4.41585586689347084490462347489, 4.58526120228103052961464181371, 5.09638095647658878650608673722, 5.38863418128718444736954436328, 5.97548267298282443053485818395, 6.43932913120193562062280154793, 6.99607528598158030816254869546, 7.35450248143790297800467637114, 7.71410731117866147437499698865, 8.277874377956488697303012697109