| L(s) = 1 | + 2·3-s − 4·7-s + 9-s + 4·13-s + 8·19-s − 8·21-s + 25-s − 4·27-s + 8·31-s + 4·37-s + 8·39-s + 20·43-s − 2·49-s + 16·57-s + 4·61-s − 4·63-s − 4·67-s + 4·73-s + 2·75-s − 16·79-s − 11·81-s − 16·91-s + 16·93-s + 4·97-s − 28·103-s + 4·109-s + 8·111-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s + 1/3·9-s + 1.10·13-s + 1.83·19-s − 1.74·21-s + 1/5·25-s − 0.769·27-s + 1.43·31-s + 0.657·37-s + 1.28·39-s + 3.04·43-s − 2/7·49-s + 2.11·57-s + 0.512·61-s − 0.503·63-s − 0.488·67-s + 0.468·73-s + 0.230·75-s − 1.80·79-s − 1.22·81-s − 1.67·91-s + 1.65·93-s + 0.406·97-s − 2.75·103-s + 0.383·109-s + 0.759·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.854188003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.854188003\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.683565298540966130897522415203, −9.633261704500415779063377753491, −8.847850276232423443211930888771, −8.740293324354028093320034492248, −7.74818737522785607031159368119, −7.71273110823499542846308181544, −6.88656092970954199609923322632, −6.27087624192875571051851265258, −5.90138678053477774939394585997, −5.17606419420266207753384183426, −4.12250433368686324236236368171, −3.70610300459195515417889608627, −2.84705552122166969163585152299, −2.76929890617261215013507568311, −1.16915866227454488376692012454,
1.16915866227454488376692012454, 2.76929890617261215013507568311, 2.84705552122166969163585152299, 3.70610300459195515417889608627, 4.12250433368686324236236368171, 5.17606419420266207753384183426, 5.90138678053477774939394585997, 6.27087624192875571051851265258, 6.88656092970954199609923322632, 7.71273110823499542846308181544, 7.74818737522785607031159368119, 8.740293324354028093320034492248, 8.847850276232423443211930888771, 9.633261704500415779063377753491, 9.683565298540966130897522415203
