| L(s) = 1 | + 1.36·3-s + 2.86·5-s − 1.14·9-s − 4.23·11-s − 13-s + 3.90·15-s − 4.41·17-s − 1.77·19-s − 5.64·23-s + 3.20·25-s − 5.64·27-s − 1.21·29-s + 10.0·31-s − 5.78·33-s − 8.72·37-s − 1.36·39-s + 8.58·41-s + 2.08·43-s − 3.26·45-s + 11.1·47-s − 6.02·51-s − 12.8·53-s − 12.1·55-s − 2.42·57-s − 4.92·59-s − 10.0·61-s − 2.86·65-s + ⋯ |
| L(s) = 1 | + 0.787·3-s + 1.28·5-s − 0.380·9-s − 1.27·11-s − 0.277·13-s + 1.00·15-s − 1.07·17-s − 0.407·19-s − 1.17·23-s + 0.641·25-s − 1.08·27-s − 0.225·29-s + 1.81·31-s − 1.00·33-s − 1.43·37-s − 0.218·39-s + 1.34·41-s + 0.318·43-s − 0.486·45-s + 1.62·47-s − 0.843·51-s − 1.76·53-s − 1.63·55-s − 0.320·57-s − 0.641·59-s − 1.28·61-s − 0.355·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 1.77T + 19T^{2} \) |
| 23 | \( 1 + 5.64T + 23T^{2} \) |
| 29 | \( 1 + 1.21T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 8.72T + 37T^{2} \) |
| 41 | \( 1 - 8.58T + 41T^{2} \) |
| 43 | \( 1 - 2.08T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 4.92T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 0.0954T + 67T^{2} \) |
| 71 | \( 1 + 8.21T + 71T^{2} \) |
| 73 | \( 1 + 0.961T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 1.66T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−7.989506073500568982494016029028, −7.28532246941740559811895348106, −6.18352466056512596732039098942, −5.87772681114665156398046380321, −4.94960529240010663368439908873, −4.17011803583241425659217165604, −2.89696087476953286631542788262, −2.47336201040180894111819122505, −1.75896440846401849708048600768, 0,
1.75896440846401849708048600768, 2.47336201040180894111819122505, 2.89696087476953286631542788262, 4.17011803583241425659217165604, 4.94960529240010663368439908873, 5.87772681114665156398046380321, 6.18352466056512596732039098942, 7.28532246941740559811895348106, 7.989506073500568982494016029028
