| L(s) = 1 | + 3-s + 9-s + 3·11-s − 3·17-s + 25-s + 27-s + 3·33-s − 3·41-s − 12·43-s − 2·49-s − 3·51-s + 6·59-s − 18·67-s − 13·73-s + 75-s + 81-s + 3·89-s − 15·97-s + 3·99-s + 3·107-s + 3·113-s − 3·121-s − 3·123-s + 127-s − 12·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.904·11-s − 0.727·17-s + 1/5·25-s + 0.192·27-s + 0.522·33-s − 0.468·41-s − 1.82·43-s − 2/7·49-s − 0.420·51-s + 0.781·59-s − 2.19·67-s − 1.52·73-s + 0.115·75-s + 1/9·81-s + 0.317·89-s − 1.52·97-s + 0.301·99-s + 0.290·107-s + 0.282·113-s − 0.272·121-s − 0.270·123-s + 0.0887·127-s − 1.05·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−7.42640598083528495735884427028, −7.13300870988558912015038793249, −6.72250537167912904301811325692, −6.35859463272807462699065386645, −5.90482393449345899968431475796, −5.33480966033838790214218804185, −4.83276037752517891929688987347, −4.36564341859125572602177028104, −4.02512598288478350875485423654, −3.35450518969791974039226032984, −3.06109551670123444543846987822, −2.35388922965370372388656964112, −1.73160637964822967975216202755, −1.23011882348805716680115620426, 0,
1.23011882348805716680115620426, 1.73160637964822967975216202755, 2.35388922965370372388656964112, 3.06109551670123444543846987822, 3.35450518969791974039226032984, 4.02512598288478350875485423654, 4.36564341859125572602177028104, 4.83276037752517891929688987347, 5.33480966033838790214218804185, 5.90482393449345899968431475796, 6.35859463272807462699065386645, 6.72250537167912904301811325692, 7.13300870988558912015038793249, 7.42640598083528495735884427028
