L(s) = 1 | − 3-s + 5-s − 7-s − 2·9-s − 11-s − 13-s − 15-s + 2·17-s − 19-s + 21-s + 25-s + 5·27-s + 6·29-s + 33-s − 35-s + 4·37-s + 39-s − 2·41-s − 2·45-s − 6·49-s − 2·51-s + 2·53-s − 55-s + 57-s − 3·59-s + 14·61-s + 2·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.277·13-s − 0.258·15-s + 0.485·17-s − 0.229·19-s + 0.218·21-s + 1/5·25-s + 0.962·27-s + 1.11·29-s + 0.174·33-s − 0.169·35-s + 0.657·37-s + 0.160·39-s − 0.312·41-s − 0.298·45-s − 6/7·49-s − 0.280·51-s + 0.274·53-s − 0.134·55-s + 0.132·57-s − 0.390·59-s + 1.79·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77932244040255403947640205964, −6.75582306573438313665376479569, −6.34302931134054931805233737710, −5.52760641360161750962859992703, −5.07006100942326275278962315301, −4.12496250074832334144634746085, −3.05550757727067772325746999218, −2.46388802151430345528390776161, −1.18107094186407499123959466774, 0,
1.18107094186407499123959466774, 2.46388802151430345528390776161, 3.05550757727067772325746999218, 4.12496250074832334144634746085, 5.07006100942326275278962315301, 5.52760641360161750962859992703, 6.34302931134054931805233737710, 6.75582306573438313665376479569, 7.77932244040255403947640205964