L(s) = 1 | + 2·3-s + 3·9-s − 2·11-s − 2·17-s − 2·19-s − 2·25-s + 4·27-s − 4·33-s − 2·41-s + 4·49-s − 4·51-s − 4·57-s − 10·59-s − 10·67-s + 2·73-s − 4·75-s + 5·81-s + 10·83-s + 20·97-s − 6·99-s − 10·107-s + 20·113-s − 18·121-s − 4·123-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 0.603·11-s − 0.485·17-s − 0.458·19-s − 2/5·25-s + 0.769·27-s − 0.696·33-s − 0.312·41-s + 4/7·49-s − 0.560·51-s − 0.529·57-s − 1.30·59-s − 1.22·67-s + 0.234·73-s − 0.461·75-s + 5/9·81-s + 1.09·83-s + 2.03·97-s − 0.603·99-s − 0.966·107-s + 1.88·113-s − 1.63·121-s − 0.360·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.59959876947910569684670698561, −7.04199868862224745079534710431, −6.68026581516752410080659109180, −6.17248148403944551757765625316, −5.81217950442415499467179051271, −5.14620866012404508676797527792, −4.75886145358235576813375766680, −4.32711587907875154540884220815, −3.81019427648662457816539581677, −3.36081273364385627024990085965, −2.85187120849606183914853523407, −2.31847701745463138788255012871, −1.92649056710611057380711780535, −1.15452478260555850561970752862, 0,
1.15452478260555850561970752862, 1.92649056710611057380711780535, 2.31847701745463138788255012871, 2.85187120849606183914853523407, 3.36081273364385627024990085965, 3.81019427648662457816539581677, 4.32711587907875154540884220815, 4.75886145358235576813375766680, 5.14620866012404508676797527792, 5.81217950442415499467179051271, 6.17248148403944551757765625316, 6.68026581516752410080659109180, 7.04199868862224745079534710431, 7.59959876947910569684670698561