| L(s) = 1 | + 3-s + 3·5-s − 2·9-s + 3·15-s − 9·23-s + 4·25-s − 5·27-s − 5·31-s − 7·37-s − 6·45-s − 12·47-s − 7·49-s − 6·53-s + 15·59-s − 13·67-s − 9·69-s − 3·71-s + 4·75-s + 81-s − 9·89-s − 5·93-s + 17·97-s − 4·103-s − 7·111-s + 21·113-s − 27·115-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 2/3·9-s + 0.774·15-s − 1.87·23-s + 4/5·25-s − 0.962·27-s − 0.898·31-s − 1.15·37-s − 0.894·45-s − 1.75·47-s − 49-s − 0.824·53-s + 1.95·59-s − 1.58·67-s − 1.08·69-s − 0.356·71-s + 0.461·75-s + 1/9·81-s − 0.953·89-s − 0.518·93-s + 1.72·97-s − 0.394·103-s − 0.664·111-s + 1.97·113-s − 2.51·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.62938943795449180375726272083, −6.71384769060720120193882421718, −6.02262961007585355420168414693, −5.59062275426054906685177336722, −4.79326113227920264944219718231, −3.73664680680989614535616825308, −3.04986442603112923409841815740, −2.09284764022340766723466984716, −1.69684964061657252783071771174, 0,
1.69684964061657252783071771174, 2.09284764022340766723466984716, 3.04986442603112923409841815740, 3.73664680680989614535616825308, 4.79326113227920264944219718231, 5.59062275426054906685177336722, 6.02262961007585355420168414693, 6.71384769060720120193882421718, 7.62938943795449180375726272083
