| L(s) = 1 | + 2·3-s − 4·4-s − 3·9-s + 3·11-s − 8·12-s + 12·16-s + 12·23-s − 10·25-s − 14·27-s − 8·31-s + 6·33-s + 12·36-s + 2·37-s − 12·44-s + 6·47-s + 24·48-s − 13·49-s − 6·53-s + 24·59-s − 32·64-s − 8·67-s + 24·69-s − 30·71-s − 20·75-s − 4·81-s + 12·89-s − 48·92-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s − 9-s + 0.904·11-s − 2.30·12-s + 3·16-s + 2.50·23-s − 2·25-s − 2.69·27-s − 1.43·31-s + 1.04·33-s + 2·36-s + 0.328·37-s − 1.80·44-s + 0.875·47-s + 3.46·48-s − 1.85·49-s − 0.824·53-s + 3.12·59-s − 4·64-s − 0.977·67-s + 2.88·69-s − 3.56·71-s − 2.30·75-s − 4/9·81-s + 1.27·89-s − 5.00·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165649 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165649 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.025761745449995221183462720434, −8.788795132432404424531313566627, −8.245334087415059049744571750965, −7.59911177067371416247669368567, −7.47786910152687687837086920367, −6.41790287732224176134698660509, −5.77160718365775620928715416255, −5.44973416215471530225317007593, −4.85705795436332516583295721672, −4.20249486126753516290654770214, −3.50910294340479626549928321873, −3.43304538851236867718675799650, −2.53090503616513762979980922027, −1.37468511161744892151372453811, 0,
1.37468511161744892151372453811, 2.53090503616513762979980922027, 3.43304538851236867718675799650, 3.50910294340479626549928321873, 4.20249486126753516290654770214, 4.85705795436332516583295721672, 5.44973416215471530225317007593, 5.77160718365775620928715416255, 6.41790287732224176134698660509, 7.47786910152687687837086920367, 7.59911177067371416247669368567, 8.245334087415059049744571750965, 8.788795132432404424531313566627, 9.025761745449995221183462720434
