L(s) = 1 | − 4·3-s + 4-s − 2·5-s + 6·9-s + 6·11-s − 4·12-s + 8·15-s + 16-s − 2·20-s − 12·23-s + 3·25-s + 4·27-s + 4·31-s − 24·33-s + 6·36-s + 4·37-s + 6·44-s − 12·45-s + 24·47-s − 4·48-s − 10·49-s + 12·53-s − 12·55-s + 8·60-s + 64-s + 16·67-s + 48·69-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 1/2·4-s − 0.894·5-s + 2·9-s + 1.80·11-s − 1.15·12-s + 2.06·15-s + 1/4·16-s − 0.447·20-s − 2.50·23-s + 3/5·25-s + 0.769·27-s + 0.718·31-s − 4.17·33-s + 36-s + 0.657·37-s + 0.904·44-s − 1.78·45-s + 3.50·47-s − 0.577·48-s − 1.42·49-s + 1.64·53-s − 1.61·55-s + 1.03·60-s + 1/8·64-s + 1.95·67-s + 5.77·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3496900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3496900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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
−7.11851346414012431362820808809, −6.57031762772610855261183604923, −6.45585392337489014705408632794, −6.07426628219491482498756116972, −5.70420801442767768967158455386, −5.40589716760043374816163821566, −4.72578510363471114360841233586, −4.36578840162334134912004410125, −3.84099354294117342446149244692, −3.73800903097193862767805884366, −2.71423556899147632833679190222, −2.21145305243918809932809731644, −1.22809259267911029527381436025, −0.827444255342549639673433939905, 0,
0.827444255342549639673433939905, 1.22809259267911029527381436025, 2.21145305243918809932809731644, 2.71423556899147632833679190222, 3.73800903097193862767805884366, 3.84099354294117342446149244692, 4.36578840162334134912004410125, 4.72578510363471114360841233586, 5.40589716760043374816163821566, 5.70420801442767768967158455386, 6.07426628219491482498756116972, 6.45585392337489014705408632794, 6.57031762772610855261183604923, 7.11851346414012431362820808809