L(s) = 1 | − 2-s + 3-s + 4-s + 0.600·5-s − 6-s + 7-s − 8-s + 9-s − 0.600·10-s + 3.98·11-s + 12-s − 2.21·13-s − 14-s + 0.600·15-s + 16-s + 0.291·17-s − 18-s + 0.811·19-s + 0.600·20-s + 21-s − 3.98·22-s + 1.18·23-s − 24-s − 4.63·25-s + 2.21·26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.268·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.189·10-s + 1.20·11-s + 0.288·12-s − 0.613·13-s − 0.267·14-s + 0.155·15-s + 0.250·16-s + 0.0706·17-s − 0.235·18-s + 0.186·19-s + 0.134·20-s + 0.218·21-s − 0.849·22-s + 0.248·23-s − 0.204·24-s − 0.927·25-s + 0.433·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.183774288\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.183774288\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
good | 5 | \( 1 - 0.600T + 5T^{2} \) |
| 11 | \( 1 - 3.98T + 11T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 - 0.291T + 17T^{2} \) |
| 19 | \( 1 - 0.811T + 19T^{2} \) |
| 23 | \( 1 - 1.18T + 23T^{2} \) |
| 29 | \( 1 + 9.44T + 29T^{2} \) |
| 31 | \( 1 - 7.61T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 - 2.80T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 - 5.87T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 5.71T + 59T^{2} \) |
| 61 | \( 1 - 2.62T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 16.8T + 73T^{2} \) |
| 79 | \( 1 - 1.60T + 79T^{2} \) |
| 83 | \( 1 + 5.30T + 83T^{2} \) |
| 89 | \( 1 + 6.96T + 89T^{2} \) |
| 97 | \( 1 - 4.49T + 97T^{2} \) |
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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
−7.973512764018298707914783773536, −7.18463526924908048401317782970, −6.73728851878388505736483612966, −5.84213627780304334251468270947, −5.10698039217713558000545423768, −4.08107290777813204318542642783, −3.50098673376109388977637048792, −2.38202306883142884314677596572, −1.81212072732611530183653268522, −0.808959527546273191324002606733,
0.808959527546273191324002606733, 1.81212072732611530183653268522, 2.38202306883142884314677596572, 3.50098673376109388977637048792, 4.08107290777813204318542642783, 5.10698039217713558000545423768, 5.84213627780304334251468270947, 6.73728851878388505736483612966, 7.18463526924908048401317782970, 7.973512764018298707914783773536