| L(s) = 1 | − 2-s − 3-s − 4-s − 2·5-s + 6-s + 2·7-s + 3·8-s + 9-s + 2·10-s + 11-s + 12-s + 13-s − 2·14-s + 2·15-s − 16-s − 2·17-s − 18-s − 6·19-s + 2·20-s − 2·21-s − 22-s − 3·24-s − 25-s − 26-s − 27-s − 2·28-s + 4·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 1.37·19-s + 0.447·20-s − 0.436·21-s − 0.213·22-s − 0.612·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226941 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226941 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.21585006272493, −12.93202046093665, −12.41978403184110, −11.92747218108972, −11.40830155449940, −11.04290340245947, −10.61405148872889, −10.34049452165984, −9.589863253099106, −9.100415944599938, −8.640522826677090, −8.368752615429895, −7.711621654946390, −7.470048270550272, −6.854042535756261, −6.294272345379795, −5.743585574408418, −5.053833761601691, −4.574600023546452, −4.312557401963963, −3.714624333394144, −3.182260531649130, −1.993303291754536, −1.740518377185646, −0.9964210289951538, 0, 0,
0.9964210289951538, 1.740518377185646, 1.993303291754536, 3.182260531649130, 3.714624333394144, 4.312557401963963, 4.574600023546452, 5.053833761601691, 5.743585574408418, 6.294272345379795, 6.854042535756261, 7.470048270550272, 7.711621654946390, 8.368752615429895, 8.640522826677090, 9.100415944599938, 9.589863253099106, 10.34049452165984, 10.61405148872889, 11.04290340245947, 11.40830155449940, 11.92747218108972, 12.41978403184110, 12.93202046093665, 13.21585006272493
