| L(s) = 1 | − 3.32·3-s − 1.32·7-s + 8.02·9-s + 5.32·11-s + 5.02·13-s + 6.34·17-s − 4.34·19-s + 4.38·21-s + 1.70·23-s − 16.6·27-s − 29-s + 8.34·31-s − 17.6·33-s + 6.93·37-s − 16.6·39-s − 1.02·41-s − 10.7·43-s + 0.679·47-s − 5.25·49-s − 21.0·51-s − 2.38·53-s + 14.4·57-s + 10.4·59-s − 6.38·61-s − 10.6·63-s + 5.70·67-s − 5.67·69-s + ⋯ |
| L(s) = 1 | − 1.91·3-s − 0.499·7-s + 2.67·9-s + 1.60·11-s + 1.39·13-s + 1.53·17-s − 0.997·19-s + 0.957·21-s + 0.356·23-s − 3.21·27-s − 0.185·29-s + 1.49·31-s − 3.07·33-s + 1.14·37-s − 2.67·39-s − 0.160·41-s − 1.63·43-s + 0.0990·47-s − 0.750·49-s − 2.95·51-s − 0.327·53-s + 1.91·57-s + 1.35·59-s − 0.817·61-s − 1.33·63-s + 0.697·67-s − 0.682·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.140378157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140378157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 3.32T + 3T^{2} \) |
| 7 | \( 1 + 1.32T + 7T^{2} \) |
| 11 | \( 1 - 5.32T + 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 - 6.34T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 - 6.93T + 37T^{2} \) |
| 41 | \( 1 + 1.02T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 0.679T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 6.38T + 61T^{2} \) |
| 67 | \( 1 - 5.70T + 67T^{2} \) |
| 71 | \( 1 + 3.61T + 71T^{2} \) |
| 73 | \( 1 - 6.73T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 3.96T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−8.859947596801216698126994037217, −7.892276066634890883392269127620, −6.78554314339873632246688805881, −6.35373105138744104665898495714, −5.97252237628886648743254358695, −4.99512575925023220208106208490, −4.14302502556374491730684091335, −3.45742923256507106697390408647, −1.50299353582171850178494945310, −0.819126102241243571686664200103,
0.819126102241243571686664200103, 1.50299353582171850178494945310, 3.45742923256507106697390408647, 4.14302502556374491730684091335, 4.99512575925023220208106208490, 5.97252237628886648743254358695, 6.35373105138744104665898495714, 6.78554314339873632246688805881, 7.892276066634890883392269127620, 8.859947596801216698126994037217
