| L(s) = 1 | + 2-s − 2.60·3-s + 4-s − 3.80·5-s − 2.60·6-s + 0.487·7-s + 8-s + 3.76·9-s − 3.80·10-s + 2.94·11-s − 2.60·12-s + 1.21·13-s + 0.487·14-s + 9.90·15-s + 16-s + 3.19·17-s + 3.76·18-s − 2.02·19-s − 3.80·20-s − 1.26·21-s + 2.94·22-s + 7.20·23-s − 2.60·24-s + 9.49·25-s + 1.21·26-s − 1.98·27-s + 0.487·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.50·3-s + 0.5·4-s − 1.70·5-s − 1.06·6-s + 0.184·7-s + 0.353·8-s + 1.25·9-s − 1.20·10-s + 0.887·11-s − 0.750·12-s + 0.336·13-s + 0.130·14-s + 2.55·15-s + 0.250·16-s + 0.773·17-s + 0.886·18-s − 0.464·19-s − 0.851·20-s − 0.276·21-s + 0.627·22-s + 1.50·23-s − 0.530·24-s + 1.89·25-s + 0.238·26-s − 0.381·27-s + 0.0921·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.063365054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063365054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 + 2.60T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 - 0.487T + 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 - 7.20T + 23T^{2} \) |
| 29 | \( 1 - 3.48T + 29T^{2} \) |
| 31 | \( 1 + 5.14T + 31T^{2} \) |
| 37 | \( 1 - 1.03T + 37T^{2} \) |
| 41 | \( 1 - 8.43T + 41T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 - 4.41T + 47T^{2} \) |
| 53 | \( 1 - 1.19T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 8.87T + 71T^{2} \) |
| 73 | \( 1 - 9.05T + 73T^{2} \) |
| 79 | \( 1 + 7.41T + 79T^{2} \) |
| 83 | \( 1 - 6.48T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 2.51T + 97T^{2} \) |
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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
−11.19644030533586092061710222292, −10.54301832860961326102753022770, −9.000030284264886444970901724103, −7.80331137287371795076369539209, −6.99217284471979371273258466452, −6.18227294914225508366460331147, −5.06572268747185624020716272143, −4.31351821819993856470546674432, −3.37955444457589713494500281434, −0.931043460744811363302608519953,
0.931043460744811363302608519953, 3.37955444457589713494500281434, 4.31351821819993856470546674432, 5.06572268747185624020716272143, 6.18227294914225508366460331147, 6.99217284471979371273258466452, 7.80331137287371795076369539209, 9.000030284264886444970901724103, 10.54301832860961326102753022770, 11.19644030533586092061710222292
