L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 5.57·11-s − 12-s − 0.613·13-s − 14-s + 15-s + 16-s + 7.52·17-s − 18-s + 6.95·19-s − 20-s − 21-s − 5.57·22-s − 23-s + 24-s + 25-s + 0.613·26-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 + 1.67·11-s − 0.288·12-s − 0.170·13-s − 0.267·14-s + 0.258·15-s + 0.250·16-s + 1.82·17-s − 0.235·18-s + 1.59·19-s − 0.223·20-s − 0.218·21-s − 1.18·22-s − 0.208·23-s + 0.204·24-s + 0.200·25-s + 0.120·26-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
\(L(1)\) |
\(\approx\) |
\(1.522127979\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.522127979\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 5.57T + 11T^{2} \) |
| 13 | \( 1 + 0.613T + 13T^{2} \) |
| 17 | \( 1 - 7.52T + 17T^{2} \) |
| 19 | \( 1 - 6.95T + 19T^{2} \) |
| 29 | \( 1 - 9.81T + 29T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 - 1.67T + 37T^{2} \) |
| 41 | \( 1 + 4.56T + 41T^{2} \) |
| 43 | \( 1 - 5.89T + 43T^{2} \) |
| 47 | \( 1 + 8.18T + 47T^{2} \) |
| 53 | \( 1 + 4.23T + 53T^{2} \) |
| 59 | \( 1 - 0.324T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 - 5.24T + 67T^{2} \) |
| 71 | \( 1 + 4.66T + 71T^{2} \) |
| 73 | \( 1 - 5.46T + 73T^{2} \) |
| 79 | \( 1 - 8.90T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 0.289T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.042489941286742202654839345344, −7.78365906746709818442909723729, −6.79056378520656345655444703986, −6.34285039151506091689017712442, −5.39691806332339475250014009134, −4.64721632805663775180340909373, −3.67183601052731066589904903958, −2.91550219781864008999809296123, −1.33942069578810664478532370058, −0.948631741792337751609330400456,
0.948631741792337751609330400456, 1.33942069578810664478532370058, 2.91550219781864008999809296123, 3.67183601052731066589904903958, 4.64721632805663775180340909373, 5.39691806332339475250014009134, 6.34285039151506091689017712442, 6.79056378520656345655444703986, 7.78365906746709818442909723729, 8.042489941286742202654839345344