L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 1.19·11-s − 12-s − 5.78·13-s − 14-s + 15-s + 16-s − 7.28·17-s − 18-s − 2.59·19-s − 20-s − 21-s − 1.19·22-s − 23-s + 24-s + 25-s + 5.78·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.360·11-s − 0.288·12-s − 1.60·13-s − 0.267·14-s + 0.258·15-s + 0.250·16-s − 1.76·17-s − 0.235·18-s − 0.594·19-s − 0.223·20-s − 0.218·21-s − 0.254·22-s − 0.208·23-s + 0.204·24-s + 0.200·25-s + 1.13·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5101473493\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5101473493\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 1.19T + 11T^{2} \) |
| 13 | \( 1 + 5.78T + 13T^{2} \) |
| 17 | \( 1 + 7.28T + 17T^{2} \) |
| 19 | \( 1 + 2.59T + 19T^{2} \) |
| 29 | \( 1 + 6.74T + 29T^{2} \) |
| 31 | \( 1 - 4.65T + 31T^{2} \) |
| 37 | \( 1 + 5.24T + 37T^{2} \) |
| 41 | \( 1 - 0.694T + 41T^{2} \) |
| 43 | \( 1 - 8.43T + 43T^{2} \) |
| 47 | \( 1 + 8.98T + 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 59 | \( 1 - 7.24T + 59T^{2} \) |
| 61 | \( 1 + 0.390T + 61T^{2} \) |
| 67 | \( 1 + 6.04T + 67T^{2} \) |
| 71 | \( 1 - 3.13T + 71T^{2} \) |
| 73 | \( 1 - 3.63T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 4.20T + 83T^{2} \) |
| 89 | \( 1 - 1.45T + 89T^{2} \) |
| 97 | \( 1 - 6.64T + 97T^{2} \) |
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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
−8.278766528407662072316151169823, −7.50124959596973716736555551323, −6.94197471576448591032966778881, −6.34523398522626808039486751198, −5.32206964364274080944830151557, −4.59851805261586259200422005613, −3.90834623448913605340750787280, −2.54332172871031250367136174062, −1.86666299676375559123160895084, −0.43189114181181170815413628831,
0.43189114181181170815413628831, 1.86666299676375559123160895084, 2.54332172871031250367136174062, 3.90834623448913605340750787280, 4.59851805261586259200422005613, 5.32206964364274080944830151557, 6.34523398522626808039486751198, 6.94197471576448591032966778881, 7.50124959596973716736555551323, 8.278766528407662072316151169823