| L(s) = 1 | + 3.46·5-s − 7-s − 2·13-s − 3.46·17-s − 19-s − 4·23-s + 6.99·25-s + 7.46·29-s − 2.92·31-s − 3.46·35-s − 2·37-s − 6·41-s − 6.92·43-s − 2.53·47-s + 49-s − 3.46·53-s − 4·59-s + 6·61-s − 6.92·65-s + 1.07·67-s − 6.53·71-s + 12.9·73-s − 1.46·83-s − 11.9·85-s − 11.8·89-s + 2·91-s − 3.46·95-s + ⋯ |
| L(s) = 1 | + 1.54·5-s − 0.377·7-s − 0.554·13-s − 0.840·17-s − 0.229·19-s − 0.834·23-s + 1.39·25-s + 1.38·29-s − 0.525·31-s − 0.585·35-s − 0.328·37-s − 0.937·41-s − 1.05·43-s − 0.369·47-s + 0.142·49-s − 0.475·53-s − 0.520·59-s + 0.768·61-s − 0.859·65-s + 0.130·67-s − 0.775·71-s + 1.51·73-s − 0.160·83-s − 1.30·85-s − 1.25·89-s + 0.209·91-s − 0.355·95-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 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 7.46T + 29T^{2} \) |
| 31 | \( 1 + 2.92T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 + 6.53T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 1.46T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.12280852183328689774986411995, −6.52138914949929911361594868312, −6.14330914560680280850677665193, −5.26134955269787280389645544576, −4.79195791719981943224983072364, −3.79119547248927746536186298081, −2.80952444463012360969730288573, −2.18169991225806667497019263958, −1.44140656629989608823028780018, 0,
1.44140656629989608823028780018, 2.18169991225806667497019263958, 2.80952444463012360969730288573, 3.79119547248927746536186298081, 4.79195791719981943224983072364, 5.26134955269787280389645544576, 6.14330914560680280850677665193, 6.52138914949929911361594868312, 7.12280852183328689774986411995
