| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2.60·7-s − 8-s + 9-s + 10-s − 12-s − 2.60·14-s + 15-s + 16-s + 2.60·17-s − 18-s − 2.60·19-s − 20-s − 2.60·21-s + 8.60·23-s + 24-s + 25-s − 27-s + 2.60·28-s − 2.60·29-s − 30-s − 6·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.984·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.288·12-s − 0.696·14-s + 0.258·15-s + 0.250·16-s + 0.631·17-s − 0.235·18-s − 0.597·19-s − 0.223·20-s − 0.568·21-s + 1.79·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.492·28-s − 0.483·29-s − 0.182·30-s − 1.07·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.161973241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.161973241\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 - 8.60T + 23T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 5.21T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 5.21T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 5.21T + 71T^{2} \) |
| 73 | \( 1 + 8.60T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 0.788T + 89T^{2} \) |
| 97 | \( 1 + 8.60T + 97T^{2} \) |
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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
−8.233516116203643595922153958069, −7.43802972025700133762780566198, −7.11442299356434811442620883473, −6.07889858438807361810368082744, −5.36446665247032576215035983099, −4.64265289442203686894030680951, −3.77239467256028936931859240846, −2.68718468900523067124738842304, −1.60071066568241069379177313191, −0.71142162776589718656331298210,
0.71142162776589718656331298210, 1.60071066568241069379177313191, 2.68718468900523067124738842304, 3.77239467256028936931859240846, 4.64265289442203686894030680951, 5.36446665247032576215035983099, 6.07889858438807361810368082744, 7.11442299356434811442620883473, 7.43802972025700133762780566198, 8.233516116203643595922153958069
