| L(s) = 1 | − 2.74·2-s + 0.909·3-s + 5.52·4-s − 2.05·5-s − 2.49·6-s − 9.67·8-s − 2.17·9-s + 5.63·10-s + 1.82·11-s + 5.02·12-s + 1.37·13-s − 1.86·15-s + 15.4·16-s − 0.748·17-s + 5.96·18-s + 5.64·19-s − 11.3·20-s − 5.01·22-s − 1.44·23-s − 8.79·24-s − 0.775·25-s − 3.77·26-s − 4.70·27-s − 6.00·29-s + 5.12·30-s + 0.260·31-s − 23.1·32-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.524·3-s + 2.76·4-s − 0.919·5-s − 1.01·6-s − 3.42·8-s − 0.724·9-s + 1.78·10-s + 0.551·11-s + 1.45·12-s + 0.381·13-s − 0.482·15-s + 3.87·16-s − 0.181·17-s + 1.40·18-s + 1.29·19-s − 2.54·20-s − 1.06·22-s − 0.300·23-s − 1.79·24-s − 0.155·25-s − 0.740·26-s − 0.905·27-s − 1.11·29-s + 0.935·30-s + 0.0467·31-s − 4.09·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 - 0.909T + 3T^{2} \) |
| 5 | \( 1 + 2.05T + 5T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 - 1.37T + 13T^{2} \) |
| 17 | \( 1 + 0.748T + 17T^{2} \) |
| 19 | \( 1 - 5.64T + 19T^{2} \) |
| 23 | \( 1 + 1.44T + 23T^{2} \) |
| 29 | \( 1 + 6.00T + 29T^{2} \) |
| 31 | \( 1 - 0.260T + 31T^{2} \) |
| 37 | \( 1 - 6.41T + 37T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 9.78T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 3.14T + 59T^{2} \) |
| 61 | \( 1 - 2.93T + 61T^{2} \) |
| 67 | \( 1 - 3.93T + 67T^{2} \) |
| 71 | \( 1 - 1.53T + 71T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + 5.62T + 97T^{2} \) |
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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
−8.341083608945358306063079827108, −7.67915541982107696831889231442, −7.20492559093872507637042326186, −6.28712646809898839197109078930, −5.51824997735812467658751800217, −3.87813940973024151152338587154, −3.18452084187240726706646987361, −2.26469080356994431933460793323, −1.16441683203550510317192458981, 0,
1.16441683203550510317192458981, 2.26469080356994431933460793323, 3.18452084187240726706646987361, 3.87813940973024151152338587154, 5.51824997735812467658751800217, 6.28712646809898839197109078930, 7.20492559093872507637042326186, 7.67915541982107696831889231442, 8.341083608945358306063079827108
