L(s) = 1 | − 2·3-s + 3·9-s − 8·11-s + 2·17-s − 8·19-s − 6·25-s − 4·27-s + 16·33-s − 12·41-s − 8·43-s + 2·49-s − 4·51-s + 16·57-s + 8·59-s + 24·67-s + 20·73-s + 12·75-s + 5·81-s − 8·83-s − 12·89-s − 12·97-s − 24·99-s − 40·107-s − 12·113-s + 26·121-s + 24·123-s + 127-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s − 2.41·11-s + 0.485·17-s − 1.83·19-s − 6/5·25-s − 0.769·27-s + 2.78·33-s − 1.87·41-s − 1.21·43-s + 2/7·49-s − 0.560·51-s + 2.11·57-s + 1.04·59-s + 2.93·67-s + 2.34·73-s + 1.38·75-s + 5/9·81-s − 0.878·83-s − 1.27·89-s − 1.21·97-s − 2.41·99-s − 3.86·107-s − 1.12·113-s + 2.36·121-s + 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−6.98083747403506855455974764057, −6.62202948008203069684059614968, −6.59291711794167281790003444711, −5.67636732029647007076099939422, −5.51324452492528964790663180164, −5.13001221098195863338015289116, −4.91029437030238646203911839660, −4.02556354189723006943550620116, −3.92247845698833805317728267090, −3.12397148446932752142076853180, −2.35058397621900011908829435300, −2.18463537719495235457264747710, −1.26926232847077019067068429617, 0, 0,
1.26926232847077019067068429617, 2.18463537719495235457264747710, 2.35058397621900011908829435300, 3.12397148446932752142076853180, 3.92247845698833805317728267090, 4.02556354189723006943550620116, 4.91029437030238646203911839660, 5.13001221098195863338015289116, 5.51324452492528964790663180164, 5.67636732029647007076099939422, 6.59291711794167281790003444711, 6.62202948008203069684059614968, 6.98083747403506855455974764057