| L(s) = 1 | + 2·3-s + 3·9-s − 8·11-s − 4·17-s + 8·19-s − 10·25-s + 4·27-s − 16·33-s + 12·41-s − 8·43-s + 2·49-s − 8·51-s + 16·57-s + 24·59-s − 24·67-s − 12·73-s − 20·75-s + 5·81-s + 8·83-s − 12·89-s − 4·97-s − 24·99-s − 8·107-s − 28·113-s + 26·121-s + 24·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 2.41·11-s − 0.970·17-s + 1.83·19-s − 2·25-s + 0.769·27-s − 2.78·33-s + 1.87·41-s − 1.21·43-s + 2/7·49-s − 1.12·51-s + 2.11·57-s + 3.12·59-s − 2.93·67-s − 1.40·73-s − 2.30·75-s + 5/9·81-s + 0.878·83-s − 1.27·89-s − 0.406·97-s − 2.41·99-s − 0.773·107-s − 2.63·113-s + 2.36·121-s + 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.069916096323518080594124819659, −7.84363031960006724677694678337, −7.42165581773296115004203753691, −7.15944714304425705232693435926, −6.43663656032022884828664728498, −5.67206581946498649675335111392, −5.47500942227311527725009815404, −4.95203593728251161575038294540, −4.25300875683489800803673652068, −3.84720863319685552465873529850, −3.10975929333988114178652450581, −2.59900951660796699369904554010, −2.33855546999761743371608352814, −1.39752451983425368394719286578, 0,
1.39752451983425368394719286578, 2.33855546999761743371608352814, 2.59900951660796699369904554010, 3.10975929333988114178652450581, 3.84720863319685552465873529850, 4.25300875683489800803673652068, 4.95203593728251161575038294540, 5.47500942227311527725009815404, 5.67206581946498649675335111392, 6.43663656032022884828664728498, 7.15944714304425705232693435926, 7.42165581773296115004203753691, 7.84363031960006724677694678337, 8.069916096323518080594124819659
