| L(s) = 1 | + 0.732·5-s + 7-s + 1.46·13-s + 4.73·17-s − 19-s − 4.46·25-s − 7.66·29-s + 2·31-s + 0.732·35-s − 10·37-s − 4.92·41-s − 8.92·43-s − 5.66·47-s + 49-s + 3.26·53-s − 8·59-s − 2·61-s + 1.07·65-s − 6.53·67-s − 1.26·71-s − 10.3·73-s + 4·79-s + 0.196·83-s + 3.46·85-s + 8.92·89-s + 1.46·91-s − 0.732·95-s + ⋯ |
| L(s) = 1 | + 0.327·5-s + 0.377·7-s + 0.406·13-s + 1.14·17-s − 0.229·19-s − 0.892·25-s − 1.42·29-s + 0.359·31-s + 0.123·35-s − 1.64·37-s − 0.769·41-s − 1.36·43-s − 0.825·47-s + 0.142·49-s + 0.448·53-s − 1.04·59-s − 0.256·61-s + 0.132·65-s − 0.798·67-s − 0.150·71-s − 1.21·73-s + 0.450·79-s + 0.0215·83-s + 0.375·85-s + 0.946·89-s + 0.153·91-s − 0.0751·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 + 5.66T + 47T^{2} \) |
| 53 | \( 1 - 3.26T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 6.53T + 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 0.196T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.41785725743517506545282893079, −6.65533759422408064739965805269, −5.89171176959547751169034298926, −5.37242293718513507797960455937, −4.65559354935670734794959286934, −3.68216947844877694334366524326, −3.18578486152986762121440700673, −1.95838419764699571518077665972, −1.43474684391959216273470191158, 0,
1.43474684391959216273470191158, 1.95838419764699571518077665972, 3.18578486152986762121440700673, 3.68216947844877694334366524326, 4.65559354935670734794959286934, 5.37242293718513507797960455937, 5.89171176959547751169034298926, 6.65533759422408064739965805269, 7.41785725743517506545282893079
