| L(s) = 1 | + 2·4-s + 6·7-s − 3·9-s + 3·13-s + 8·19-s − 10·25-s + 12·28-s − 15·31-s − 6·36-s − 10·37-s + 17·49-s + 6·52-s − 15·61-s − 18·63-s − 8·64-s − 5·67-s + 17·73-s + 16·76-s + 9·79-s + 9·81-s + 18·91-s − 9·97-s − 20·100-s + 20·103-s − 9·117-s + 11·121-s − 30·124-s + ⋯ |
| L(s) = 1 | + 4-s + 2.26·7-s − 9-s + 0.832·13-s + 1.83·19-s − 2·25-s + 2.26·28-s − 2.69·31-s − 36-s − 1.64·37-s + 17/7·49-s + 0.832·52-s − 1.92·61-s − 2.26·63-s − 64-s − 0.610·67-s + 1.98·73-s + 1.83·76-s + 1.01·79-s + 81-s + 1.88·91-s − 0.913·97-s − 2·100-s + 1.97·103-s − 0.832·117-s + 121-s − 2.69·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40401 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.951229600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.951229600\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 67 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−12.28451661759237724100706729835, −12.12963089637884718109286515957, −11.42294118837224217405332456831, −11.38189293323274019619792288494, −10.96024311425129136224749613788, −10.61447750941184741502720916916, −9.727610855142279157051835499105, −9.079871547189229652543877367797, −8.718983411824292301617037917654, −8.015363630144611141502948401274, −7.58034432943335284601240501042, −7.41564914703290326250863712393, −6.47848823344714132384351381309, −5.71926691402001289613629893767, −5.41922484150236154904235423533, −4.86650631054172030737067667953, −3.83491271730638232846916347580, −3.23236307143343293489171197126, −1.99469443071340221084861994500, −1.66887642201510692538169405140,
1.66887642201510692538169405140, 1.99469443071340221084861994500, 3.23236307143343293489171197126, 3.83491271730638232846916347580, 4.86650631054172030737067667953, 5.41922484150236154904235423533, 5.71926691402001289613629893767, 6.47848823344714132384351381309, 7.41564914703290326250863712393, 7.58034432943335284601240501042, 8.015363630144611141502948401274, 8.718983411824292301617037917654, 9.079871547189229652543877367797, 9.727610855142279157051835499105, 10.61447750941184741502720916916, 10.96024311425129136224749613788, 11.38189293323274019619792288494, 11.42294118837224217405332456831, 12.12963089637884718109286515957, 12.28451661759237724100706729835
