| L(s) = 1 | + 3·3-s + 5-s − 2·7-s + 4·9-s − 11-s + 13-s + 3·15-s − 12·17-s + 4·19-s − 6·21-s + 3·23-s − 6·25-s + 6·27-s − 3·29-s − 3·31-s − 3·33-s − 2·35-s − 2·37-s + 3·39-s + 9·41-s − 6·43-s + 4·45-s − 2·47-s + 2·49-s − 36·51-s + 12·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 0.755·7-s + 4/3·9-s − 0.301·11-s + 0.277·13-s + 0.774·15-s − 2.91·17-s + 0.917·19-s − 1.30·21-s + 0.625·23-s − 6/5·25-s + 1.15·27-s − 0.557·29-s − 0.538·31-s − 0.522·33-s − 0.338·35-s − 0.328·37-s + 0.480·39-s + 1.40·41-s − 0.914·43-s + 0.596·45-s − 0.291·47-s + 2/7·49-s − 5.04·51-s + 1.64·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5607424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5607424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.251242536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.251242536\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_4$ | \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 61 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 11 T + 83 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 21 T + 253 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 11 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.069209802411176238199506216307, −8.855820573373444028373712414413, −8.380031259325216394423208543422, −8.336611144699361093517820113497, −7.64429795463427911698833468025, −7.21671162425764860760013864727, −6.95825527448039212767536842635, −6.67328312580759042489787592924, −5.97357340456275159642676498569, −5.84019526948162706012289293039, −5.13168222989071166574785713754, −4.74679733400694768212541219378, −4.14541829434390951103105683313, −3.79424579121823740375501171518, −3.42950928286374688138209739524, −2.83802902324237826115557032885, −2.33349610642909562115570568698, −2.27347514723706435329817268742, −1.55299958308258776591813379532, −0.51328978480074329868698016162,
0.51328978480074329868698016162, 1.55299958308258776591813379532, 2.27347514723706435329817268742, 2.33349610642909562115570568698, 2.83802902324237826115557032885, 3.42950928286374688138209739524, 3.79424579121823740375501171518, 4.14541829434390951103105683313, 4.74679733400694768212541219378, 5.13168222989071166574785713754, 5.84019526948162706012289293039, 5.97357340456275159642676498569, 6.67328312580759042489787592924, 6.95825527448039212767536842635, 7.21671162425764860760013864727, 7.64429795463427911698833468025, 8.336611144699361093517820113497, 8.380031259325216394423208543422, 8.855820573373444028373712414413, 9.069209802411176238199506216307
