| L(s) = 1 | + 3.56·5-s − 7-s − 3.12·11-s + 13-s + 1.12·17-s − 1.56·19-s − 8.68·23-s + 7.68·25-s − 0.438·29-s − 5.56·31-s − 3.56·35-s + 1.12·37-s + 2·41-s − 9.56·43-s − 9.56·47-s + 49-s − 6.68·53-s − 11.1·55-s + 8·59-s − 2·61-s + 3.56·65-s − 7.12·67-s + 10.2·71-s − 8.43·73-s + 3.12·77-s + 8.68·79-s + 10.4·83-s + ⋯ |
| L(s) = 1 | + 1.59·5-s − 0.377·7-s − 0.941·11-s + 0.277·13-s + 0.272·17-s − 0.358·19-s − 1.81·23-s + 1.53·25-s − 0.0814·29-s − 0.998·31-s − 0.602·35-s + 0.184·37-s + 0.312·41-s − 1.45·43-s − 1.39·47-s + 0.142·49-s − 0.918·53-s − 1.49·55-s + 1.04·59-s − 0.256·61-s + 0.441·65-s − 0.870·67-s + 1.21·71-s − 0.987·73-s + 0.355·77-s + 0.977·79-s + 1.14·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 23 | \( 1 + 8.68T + 23T^{2} \) |
| 29 | \( 1 + 0.438T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 9.56T + 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 + 6.68T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 7.12T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 8.43T + 73T^{2} \) |
| 79 | \( 1 - 8.68T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 + 6.68T + 97T^{2} \) |
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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
−7.76637239056065755281555838221, −6.69679172440800416587804732072, −6.23111723969295222987770984777, −5.55407839979654207077087147523, −5.05410699077309849918930569286, −3.96090910051573456201952195769, −3.03649761434942615382834913829, −2.19172977117506031332803069858, −1.57110979582852467478554247995, 0,
1.57110979582852467478554247995, 2.19172977117506031332803069858, 3.03649761434942615382834913829, 3.96090910051573456201952195769, 5.05410699077309849918930569286, 5.55407839979654207077087147523, 6.23111723969295222987770984777, 6.69679172440800416587804732072, 7.76637239056065755281555838221
