| L(s) = 1 | + 2·3-s − 2·7-s + 9-s − 8·13-s − 4·19-s − 4·21-s − 10·25-s − 4·27-s + 8·31-s + 4·37-s − 16·39-s − 16·43-s + 3·49-s − 8·57-s + 16·61-s − 2·63-s + 8·67-s + 4·73-s − 20·75-s − 16·79-s − 11·81-s + 16·91-s + 16·93-s − 20·97-s + 8·103-s + 4·109-s + 8·111-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 1/3·9-s − 2.21·13-s − 0.917·19-s − 0.872·21-s − 2·25-s − 0.769·27-s + 1.43·31-s + 0.657·37-s − 2.56·39-s − 2.43·43-s + 3/7·49-s − 1.05·57-s + 2.04·61-s − 0.251·63-s + 0.977·67-s + 0.468·73-s − 2.30·75-s − 1.80·79-s − 1.22·81-s + 1.67·91-s + 1.65·93-s − 2.03·97-s + 0.788·103-s + 0.383·109-s + 0.759·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.392494157732895222790142944171, −8.686268182491541384374530740513, −8.182643116839158569055135937831, −7.967527986115067076241988284772, −7.29668061464756137953129595214, −6.84458804347565715458286520604, −6.34752255234086455771568318662, −5.62168100149868099368912779374, −5.05202336643031434355439952711, −4.32309265725798888275246837851, −3.82418745638366191622617161116, −3.04185374086143547449059084416, −2.48514181668809031419660323366, −1.98782431119292345428047370492, 0,
1.98782431119292345428047370492, 2.48514181668809031419660323366, 3.04185374086143547449059084416, 3.82418745638366191622617161116, 4.32309265725798888275246837851, 5.05202336643031434355439952711, 5.62168100149868099368912779374, 6.34752255234086455771568318662, 6.84458804347565715458286520604, 7.29668061464756137953129595214, 7.967527986115067076241988284772, 8.182643116839158569055135937831, 8.686268182491541384374530740513, 9.392494157732895222790142944171
