| L(s) = 1 | − 3·4-s + 2·5-s − 8·11-s + 5·16-s + 8·19-s − 6·20-s − 25-s + 4·29-s − 4·41-s + 24·44-s + 49-s − 16·55-s − 24·59-s − 4·61-s − 3·64-s − 24·76-s − 32·79-s + 10·80-s + 28·89-s + 16·95-s + 3·100-s − 28·101-s − 36·109-s − 12·116-s + 26·121-s − 12·125-s + 127-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.894·5-s − 2.41·11-s + 5/4·16-s + 1.83·19-s − 1.34·20-s − 1/5·25-s + 0.742·29-s − 0.624·41-s + 3.61·44-s + 1/7·49-s − 2.15·55-s − 3.12·59-s − 0.512·61-s − 3/8·64-s − 2.75·76-s − 3.60·79-s + 1.11·80-s + 2.96·89-s + 1.64·95-s + 3/10·100-s − 2.78·101-s − 3.44·109-s − 1.11·116-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.457705101580370517739336150901, −8.941556964198447272384176621072, −8.352294990335304362757323779032, −7.81295430367747747098376616892, −7.65494363024181651773191318169, −6.85360812870284917864517668882, −5.94607665445287604157300008429, −5.61533530390354367379245545667, −5.02709416296405066539370067618, −4.87983646890892800864940889103, −4.03796996879678948025961788837, −3.05422074105458389777226041971, −2.72775334283054769167882483900, −1.46036150187712502072371434143, 0,
1.46036150187712502072371434143, 2.72775334283054769167882483900, 3.05422074105458389777226041971, 4.03796996879678948025961788837, 4.87983646890892800864940889103, 5.02709416296405066539370067618, 5.61533530390354367379245545667, 5.94607665445287604157300008429, 6.85360812870284917864517668882, 7.65494363024181651773191318169, 7.81295430367747747098376616892, 8.352294990335304362757323779032, 8.941556964198447272384176621072, 9.457705101580370517739336150901
