| L(s) = 1 | + 2.75·2-s − 3.02·3-s + 5.58·4-s − 3.63·5-s − 8.32·6-s + 2.50·7-s + 9.88·8-s + 6.13·9-s − 10.0·10-s − 11-s − 16.8·12-s − 2.06·13-s + 6.89·14-s + 10.9·15-s + 16.0·16-s + 3.53·17-s + 16.8·18-s − 0.848·19-s − 20.3·20-s − 7.56·21-s − 2.75·22-s − 0.787·23-s − 29.8·24-s + 8.19·25-s − 5.69·26-s − 9.47·27-s + 13.9·28-s + ⋯ |
| L(s) = 1 | + 1.94·2-s − 1.74·3-s + 2.79·4-s − 1.62·5-s − 3.39·6-s + 0.946·7-s + 3.49·8-s + 2.04·9-s − 3.16·10-s − 0.301·11-s − 4.87·12-s − 0.573·13-s + 1.84·14-s + 2.83·15-s + 4.01·16-s + 0.857·17-s + 3.98·18-s − 0.194·19-s − 4.54·20-s − 1.65·21-s − 0.587·22-s − 0.164·23-s − 6.09·24-s + 1.63·25-s − 1.11·26-s − 1.82·27-s + 2.64·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.943962501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.943962501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 - 2.75T + 2T^{2} \) |
| 3 | \( 1 + 3.02T + 3T^{2} \) |
| 5 | \( 1 + 3.63T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 - 3.53T + 17T^{2} \) |
| 19 | \( 1 + 0.848T + 19T^{2} \) |
| 23 | \( 1 + 0.787T + 23T^{2} \) |
| 29 | \( 1 - 9.43T + 29T^{2} \) |
| 31 | \( 1 - 7.50T + 31T^{2} \) |
| 37 | \( 1 + 9.22T + 37T^{2} \) |
| 41 | \( 1 - 8.85T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 4.17T + 47T^{2} \) |
| 53 | \( 1 - 2.97T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 1.10T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 4.97T + 71T^{2} \) |
| 73 | \( 1 + 5.08T + 73T^{2} \) |
| 79 | \( 1 + 6.33T + 79T^{2} \) |
| 83 | \( 1 - 0.754T + 83T^{2} \) |
| 89 | \( 1 + 1.97T + 89T^{2} \) |
| 97 | \( 1 - 3.88T + 97T^{2} \) |
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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
−10.42783059607437144163274749535, −8.133463082632343894663678166716, −7.45911956001116675046941478163, −6.86389094797152205567407255756, −5.90583043488633484895832851715, −5.17466621953006309948659398243, −4.50762310448393695370301985186, −4.11940775301115082646667085499, −2.78702050024450665872763438005, −1.06721291771568276281075102681,
1.06721291771568276281075102681, 2.78702050024450665872763438005, 4.11940775301115082646667085499, 4.50762310448393695370301985186, 5.17466621953006309948659398243, 5.90583043488633484895832851715, 6.86389094797152205567407255756, 7.45911956001116675046941478163, 8.133463082632343894663678166716, 10.42783059607437144163274749535
