| L(s) = 1 | − 1.27·2-s + 1.16·3-s − 0.370·4-s − 1.81·5-s − 1.49·6-s + 3.02·8-s − 1.63·9-s + 2.31·10-s − 2.77·11-s − 0.432·12-s − 2.11·15-s − 3.12·16-s − 2.74·17-s + 2.08·18-s + 5.86·19-s + 0.672·20-s + 3.54·22-s + 6.99·23-s + 3.53·24-s − 1.70·25-s − 5.41·27-s − 3.51·29-s + 2.70·30-s + 2.06·31-s − 2.06·32-s − 3.24·33-s + 3.50·34-s + ⋯ |
| L(s) = 1 | − 0.902·2-s + 0.674·3-s − 0.185·4-s − 0.811·5-s − 0.608·6-s + 1.06·8-s − 0.545·9-s + 0.732·10-s − 0.837·11-s − 0.124·12-s − 0.547·15-s − 0.780·16-s − 0.665·17-s + 0.492·18-s + 1.34·19-s + 0.150·20-s + 0.756·22-s + 1.45·23-s + 0.721·24-s − 0.341·25-s − 1.04·27-s − 0.652·29-s + 0.494·30-s + 0.371·31-s − 0.365·32-s − 0.564·33-s + 0.600·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 3 | \( 1 - 1.16T + 3T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 11 | \( 1 + 2.77T + 11T^{2} \) |
| 17 | \( 1 + 2.74T + 17T^{2} \) |
| 19 | \( 1 - 5.86T + 19T^{2} \) |
| 23 | \( 1 - 6.99T + 23T^{2} \) |
| 29 | \( 1 + 3.51T + 29T^{2} \) |
| 31 | \( 1 - 2.06T + 31T^{2} \) |
| 37 | \( 1 - 1.74T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 - 9.10T + 43T^{2} \) |
| 47 | \( 1 + 6.65T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 3.07T + 59T^{2} \) |
| 61 | \( 1 + 1.08T + 61T^{2} \) |
| 67 | \( 1 + 5.01T + 67T^{2} \) |
| 71 | \( 1 - 2.71T + 71T^{2} \) |
| 73 | \( 1 - 7.67T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 7.97T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.62679074367848954272798036279, −7.35090663123646918422054289687, −6.17899982432536589760310479569, −5.22447678115490267362097030049, −4.65799323614488132654451836415, −3.73381152841527184954121608793, −3.04437055063035114085849777545, −2.21474658263017100251035279159, −0.984467802406303391244660972480, 0,
0.984467802406303391244660972480, 2.21474658263017100251035279159, 3.04437055063035114085849777545, 3.73381152841527184954121608793, 4.65799323614488132654451836415, 5.22447678115490267362097030049, 6.17899982432536589760310479569, 7.35090663123646918422054289687, 7.62679074367848954272798036279
