| L(s) = 1 | − 2-s + 4-s + 2.37·7-s − 8-s + 11-s − 2·13-s − 2.37·14-s + 16-s − 4.37·17-s + 6.37·19-s − 22-s − 8.74·23-s + 2·26-s + 2.37·28-s + 4.37·29-s − 2.37·31-s − 32-s + 4.37·34-s − 3.62·37-s − 6.37·38-s − 11.4·41-s + 4·43-s + 44-s + 8.74·46-s − 8.74·47-s − 1.37·49-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.896·7-s − 0.353·8-s + 0.301·11-s − 0.554·13-s − 0.634·14-s + 0.250·16-s − 1.06·17-s + 1.46·19-s − 0.213·22-s − 1.82·23-s + 0.392·26-s + 0.448·28-s + 0.811·29-s − 0.426·31-s − 0.176·32-s + 0.749·34-s − 0.596·37-s − 1.03·38-s − 1.79·41-s + 0.609·43-s + 0.150·44-s + 1.28·46-s − 1.27·47-s − 0.196·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 + 8.74T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 + 2.37T + 31T^{2} \) |
| 37 | \( 1 + 3.62T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 8.74T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 + 7.48T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 8.74T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
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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
−8.017637585006348520227328644935, −7.27710101114682377490844291281, −6.66380799495427026102804544071, −5.74760645779859951798545448350, −4.98441891198832452769047333050, −4.19712402740535266970959542782, −3.16879138609288381824344005953, −2.12449234460290937536210726977, −1.41439500488856941593153725874, 0,
1.41439500488856941593153725874, 2.12449234460290937536210726977, 3.16879138609288381824344005953, 4.19712402740535266970959542782, 4.98441891198832452769047333050, 5.74760645779859951798545448350, 6.66380799495427026102804544071, 7.27710101114682377490844291281, 8.017637585006348520227328644935
