| L(s) = 1 | + 2-s + 2.92·3-s + 4-s − 1.62·5-s + 2.92·6-s − 3.51·7-s + 8-s + 5.57·9-s − 1.62·10-s − 4.83·11-s + 2.92·12-s − 1.79·13-s − 3.51·14-s − 4.74·15-s + 16-s − 3.72·17-s + 5.57·18-s + 1.82·19-s − 1.62·20-s − 10.2·21-s − 4.83·22-s − 2.59·23-s + 2.92·24-s − 2.36·25-s − 1.79·26-s + 7.52·27-s − 3.51·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.69·3-s + 0.5·4-s − 0.725·5-s + 1.19·6-s − 1.32·7-s + 0.353·8-s + 1.85·9-s − 0.512·10-s − 1.45·11-s + 0.845·12-s − 0.497·13-s − 0.939·14-s − 1.22·15-s + 0.250·16-s − 0.903·17-s + 1.31·18-s + 0.418·19-s − 0.362·20-s − 2.24·21-s − 1.03·22-s − 0.541·23-s + 0.597·24-s − 0.473·25-s − 0.352·26-s + 1.44·27-s − 0.664·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 2.92T + 3T^{2} \) |
| 5 | \( 1 + 1.62T + 5T^{2} \) |
| 7 | \( 1 + 3.51T + 7T^{2} \) |
| 11 | \( 1 + 4.83T + 11T^{2} \) |
| 13 | \( 1 + 1.79T + 13T^{2} \) |
| 17 | \( 1 + 3.72T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 + 2.59T + 23T^{2} \) |
| 29 | \( 1 - 2.58T + 29T^{2} \) |
| 31 | \( 1 + 4.83T + 31T^{2} \) |
| 37 | \( 1 - 12.0T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 5.36T + 43T^{2} \) |
| 47 | \( 1 - 8.43T + 47T^{2} \) |
| 53 | \( 1 + 3.00T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 8.93T + 67T^{2} \) |
| 71 | \( 1 - 4.99T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 6.15T + 83T^{2} \) |
| 89 | \( 1 + 4.23T + 89T^{2} \) |
| 97 | \( 1 + 7.13T + 97T^{2} \) |
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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
−7.908573757211319661034956837715, −7.52116861524717754051778281657, −6.78936173577233442999658667635, −5.87281269443863755712287254520, −4.76571500887288937609992907604, −4.04559778827843502391970480110, −3.24731452563328928067154598474, −2.81461930249838107705682347701, −2.02732569127384081896687735073, 0,
2.02732569127384081896687735073, 2.81461930249838107705682347701, 3.24731452563328928067154598474, 4.04559778827843502391970480110, 4.76571500887288937609992907604, 5.87281269443863755712287254520, 6.78936173577233442999658667635, 7.52116861524717754051778281657, 7.908573757211319661034956837715
