| L(s) = 1 | + 2·5-s − 4·7-s − 6·13-s + 6·17-s + 8·19-s − 25-s − 6·29-s − 8·35-s + 6·37-s − 10·41-s + 8·43-s + 9·49-s − 6·53-s − 4·59-s + 2·61-s − 12·65-s − 12·67-s + 8·71-s − 2·73-s + 4·79-s − 12·83-s + 12·85-s + 6·89-s + 24·91-s + 16·95-s + 2·97-s + 2·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 1.66·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.35·35-s + 0.986·37-s − 1.56·41-s + 1.21·43-s + 9/7·49-s − 0.824·53-s − 0.520·59-s + 0.256·61-s − 1.48·65-s − 1.46·67-s + 0.949·71-s − 0.234·73-s + 0.450·79-s − 1.31·83-s + 1.30·85-s + 0.635·89-s + 2.51·91-s + 1.64·95-s + 0.203·97-s + 0.199·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.46934429894547107908108861873, −6.75049099573056630235505373284, −5.88767631750682274656040689905, −5.54566318885565894853837933993, −4.80134103241702017404924127291, −3.59975396638775029408156508083, −3.06787219115102978612636157906, −2.35200561646137909506726785661, −1.22981163846108448959993130756, 0,
1.22981163846108448959993130756, 2.35200561646137909506726785661, 3.06787219115102978612636157906, 3.59975396638775029408156508083, 4.80134103241702017404924127291, 5.54566318885565894853837933993, 5.88767631750682274656040689905, 6.75049099573056630235505373284, 7.46934429894547107908108861873
