| L(s) = 1 | − 9-s + 8·11-s − 16·19-s − 12·29-s − 12·41-s + 14·49-s − 24·59-s − 28·61-s − 16·79-s + 81-s − 4·89-s − 8·99-s + 28·101-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 16·171-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.41·11-s − 3.67·19-s − 2.22·29-s − 1.87·41-s + 2·49-s − 3.12·59-s − 3.58·61-s − 1.80·79-s + 1/9·81-s − 0.423·89-s − 0.804·99-s + 2.78·101-s + 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 1.22·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.148999417763595037242873834441, −7.71032510569072262086488957415, −7.20954408718917121186469512149, −7.17783413321688312870870099919, −6.36133176608047715519177047249, −6.35615423258339646253996866730, −5.98817181650402084242120908660, −5.93574274819714052762172486210, −4.87662642103056694520551797063, −4.83210746621997969870832160177, −4.16413160757843991694599718276, −4.09788697308073685171324925749, −3.59876710541155682093588445546, −3.29011294053351677908585571686, −2.51633575047117498215008641202, −2.10101473529025980216811474636, −1.49785612267003571358929907306, −1.45990141441098031357572591635, 0, 0,
1.45990141441098031357572591635, 1.49785612267003571358929907306, 2.10101473529025980216811474636, 2.51633575047117498215008641202, 3.29011294053351677908585571686, 3.59876710541155682093588445546, 4.09788697308073685171324925749, 4.16413160757843991694599718276, 4.83210746621997969870832160177, 4.87662642103056694520551797063, 5.93574274819714052762172486210, 5.98817181650402084242120908660, 6.35615423258339646253996866730, 6.36133176608047715519177047249, 7.17783413321688312870870099919, 7.20954408718917121186469512149, 7.71032510569072262086488957415, 8.148999417763595037242873834441
