| L(s) = 1 | − 2-s + 0.429·3-s + 4-s − 5-s − 0.429·6-s + 2.14·7-s − 8-s − 2.81·9-s + 10-s + 11-s + 0.429·12-s − 5.97·13-s − 2.14·14-s − 0.429·15-s + 16-s − 4.12·17-s + 2.81·18-s + 4.92·19-s − 20-s + 0.919·21-s − 22-s − 2.48·23-s − 0.429·24-s + 25-s + 5.97·26-s − 2.49·27-s + 2.14·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.248·3-s + 0.5·4-s − 0.447·5-s − 0.175·6-s + 0.809·7-s − 0.353·8-s − 0.938·9-s + 0.316·10-s + 0.301·11-s + 0.124·12-s − 1.65·13-s − 0.572·14-s − 0.110·15-s + 0.250·16-s − 1.00·17-s + 0.663·18-s + 1.12·19-s − 0.223·20-s + 0.200·21-s − 0.213·22-s − 0.518·23-s − 0.0877·24-s + 0.200·25-s + 1.17·26-s − 0.480·27-s + 0.404·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.032373958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.032373958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 0.429T + 3T^{2} \) |
| 7 | \( 1 - 2.14T + 7T^{2} \) |
| 13 | \( 1 + 5.97T + 13T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 - 4.92T + 19T^{2} \) |
| 23 | \( 1 + 2.48T + 23T^{2} \) |
| 29 | \( 1 - 6.49T + 29T^{2} \) |
| 31 | \( 1 - 0.915T + 31T^{2} \) |
| 37 | \( 1 + 7.98T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 47 | \( 1 + 3.09T + 47T^{2} \) |
| 53 | \( 1 - 5.16T + 53T^{2} \) |
| 59 | \( 1 - 9.13T + 59T^{2} \) |
| 61 | \( 1 - 3.40T + 61T^{2} \) |
| 67 | \( 1 - 1.10T + 67T^{2} \) |
| 71 | \( 1 + 1.82T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 + 0.758T + 83T^{2} \) |
| 89 | \( 1 + 1.69T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.405336578861037517336724761565, −7.64491343512925407059280980006, −7.13234589763829616870812702001, −6.30080452247358202103730779037, −5.22512946319893192713055863672, −4.74196226700042276713365682394, −3.60501668450273642782925838332, −2.66070153006117023001004248165, −1.97958451054522195049729731276, −0.59506198461634821673166633670,
0.59506198461634821673166633670, 1.97958451054522195049729731276, 2.66070153006117023001004248165, 3.60501668450273642782925838332, 4.74196226700042276713365682394, 5.22512946319893192713055863672, 6.30080452247358202103730779037, 7.13234589763829616870812702001, 7.64491343512925407059280980006, 8.405336578861037517336724761565
