| L(s) = 1 | + 2·2-s + 3.34·3-s + 4·4-s + 2.71·5-s + 6.69·6-s + 8·8-s − 15.7·9-s + 5.43·10-s + 14.0·11-s + 13.3·12-s + 44.4·13-s + 9.09·15-s + 16·16-s − 107.·17-s − 31.5·18-s − 34.7·19-s + 10.8·20-s + 28.1·22-s + 23·23-s + 26.7·24-s − 117.·25-s + 88.9·26-s − 143.·27-s − 237.·29-s + 18.1·30-s − 132.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.644·3-s + 0.5·4-s + 0.242·5-s + 0.455·6-s + 0.353·8-s − 0.584·9-s + 0.171·10-s + 0.385·11-s + 0.322·12-s + 0.948·13-s + 0.156·15-s + 0.250·16-s − 1.53·17-s − 0.413·18-s − 0.419·19-s + 0.121·20-s + 0.272·22-s + 0.208·23-s + 0.227·24-s − 0.940·25-s + 0.670·26-s − 1.02·27-s − 1.52·29-s + 0.110·30-s − 0.768·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 3.34T + 27T^{2} \) |
| 5 | \( 1 - 2.71T + 125T^{2} \) |
| 11 | \( 1 - 14.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 34.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 237.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 58.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 72.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 97.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 258.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 46.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 140.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 17.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 685.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 331.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 943.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 827.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 53.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 276.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 971.T + 9.12e5T^{2} \) |
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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
−8.344775979109447336616525015039, −7.50973258221066059690800599800, −6.52776688194415953904371451983, −5.98924016781866491812530513408, −5.09383415993391420215853903416, −4.00431358244011187762390211229, −3.49329933454143081462594014410, −2.36524145039993992338508241892, −1.69322811706573966944354918287, 0,
1.69322811706573966944354918287, 2.36524145039993992338508241892, 3.49329933454143081462594014410, 4.00431358244011187762390211229, 5.09383415993391420215853903416, 5.98924016781866491812530513408, 6.52776688194415953904371451983, 7.50973258221066059690800599800, 8.344775979109447336616525015039
