L(s) = 1 | − 2-s + 3-s + 4-s + 3.21·5-s − 6-s − 0.928·7-s − 8-s + 9-s − 3.21·10-s − 0.454·11-s + 12-s + 13-s + 0.928·14-s + 3.21·15-s + 16-s + 3.97·17-s − 18-s + 1.03·19-s + 3.21·20-s − 0.928·21-s + 0.454·22-s − 8.01·23-s − 24-s + 5.33·25-s − 26-s + 27-s − 0.928·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.43·5-s − 0.408·6-s − 0.350·7-s − 0.353·8-s + 0.333·9-s − 1.01·10-s − 0.137·11-s + 0.288·12-s + 0.277·13-s + 0.248·14-s + 0.830·15-s + 0.250·16-s + 0.963·17-s − 0.235·18-s + 0.237·19-s + 0.718·20-s − 0.202·21-s + 0.0969·22-s − 1.67·23-s − 0.204·24-s + 1.06·25-s − 0.196·26-s + 0.192·27-s − 0.175·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.560619758\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.560619758\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 3.21T + 5T^{2} \) |
| 7 | \( 1 + 0.928T + 7T^{2} \) |
| 11 | \( 1 + 0.454T + 11T^{2} \) |
| 17 | \( 1 - 3.97T + 17T^{2} \) |
| 19 | \( 1 - 1.03T + 19T^{2} \) |
| 23 | \( 1 + 8.01T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 - 3.55T + 31T^{2} \) |
| 37 | \( 1 + 6.52T + 37T^{2} \) |
| 41 | \( 1 + 3.35T + 41T^{2} \) |
| 43 | \( 1 - 8.82T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 0.418T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 0.360T + 61T^{2} \) |
| 67 | \( 1 - 4.15T + 67T^{2} \) |
| 71 | \( 1 - 4.04T + 71T^{2} \) |
| 73 | \( 1 - 3.48T + 73T^{2} \) |
| 79 | \( 1 - 0.976T + 79T^{2} \) |
| 83 | \( 1 + 7.97T + 83T^{2} \) |
| 89 | \( 1 + 5.87T + 89T^{2} \) |
| 97 | \( 1 - 6.80T + 97T^{2} \) |
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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
−8.026862512868615726253465074153, −7.18044345888818923937784567273, −6.49417787809324754219679338529, −5.85149852185050335025967617537, −5.32557962354068648676335062062, −4.14640746188887830440402201788, −3.24725664466321186056268006277, −2.45791721621308128847927411529, −1.82197981365536666568429230304, −0.878055164205991727427518805918,
0.878055164205991727427518805918, 1.82197981365536666568429230304, 2.45791721621308128847927411529, 3.24725664466321186056268006277, 4.14640746188887830440402201788, 5.32557962354068648676335062062, 5.85149852185050335025967617537, 6.49417787809324754219679338529, 7.18044345888818923937784567273, 8.026862512868615726253465074153