| L(s) = 1 | − 2·3-s − 2·9-s − 10·11-s + 2·19-s − 2·25-s + 10·27-s + 20·33-s − 5·41-s − 4·43-s + 10·49-s − 4·57-s − 6·67-s − 20·73-s + 4·75-s − 5·81-s − 8·89-s + 12·97-s + 20·99-s + 8·107-s − 12·113-s + 54·121-s + 10·123-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2/3·9-s − 3.01·11-s + 0.458·19-s − 2/5·25-s + 1.92·27-s + 3.48·33-s − 0.780·41-s − 0.609·43-s + 10/7·49-s − 0.529·57-s − 0.733·67-s − 2.34·73-s + 0.461·75-s − 5/9·81-s − 0.847·89-s + 1.21·97-s + 2.01·99-s + 0.773·107-s − 1.12·113-s + 4.90·121-s + 0.901·123-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−11.15662733958893988623597716409, −10.59825324071322855057939230554, −10.39235113006262168747798579527, −9.780666496112151246286232763050, −8.763473008018823706570908594628, −8.333302521823547348457406789793, −7.67147930914584082626489197721, −7.18157266326531913255505905120, −6.14734264865620643600828997855, −5.60340733293681562859559985798, −5.27924781885721236182136302857, −4.64179799435541320220033101215, −3.17814006712154085473407665632, −2.49110577870648363746397951149, 0,
2.49110577870648363746397951149, 3.17814006712154085473407665632, 4.64179799435541320220033101215, 5.27924781885721236182136302857, 5.60340733293681562859559985798, 6.14734264865620643600828997855, 7.18157266326531913255505905120, 7.67147930914584082626489197721, 8.333302521823547348457406789793, 8.763473008018823706570908594628, 9.780666496112151246286232763050, 10.39235113006262168747798579527, 10.59825324071322855057939230554, 11.15662733958893988623597716409
