| L(s) = 1 | + 2.46·2-s − 3-s + 4.06·4-s + 1.55·5-s − 2.46·6-s − 3.08·7-s + 5.09·8-s + 9-s + 3.84·10-s + 0.965·11-s − 4.06·12-s − 0.969·13-s − 7.59·14-s − 1.55·15-s + 4.41·16-s + 17-s + 2.46·18-s − 8.30·19-s + 6.34·20-s + 3.08·21-s + 2.37·22-s − 1.19·23-s − 5.09·24-s − 2.56·25-s − 2.38·26-s − 27-s − 12.5·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.577·3-s + 2.03·4-s + 0.697·5-s − 1.00·6-s − 1.16·7-s + 1.80·8-s + 0.333·9-s + 1.21·10-s + 0.291·11-s − 1.17·12-s − 0.268·13-s − 2.02·14-s − 0.402·15-s + 1.10·16-s + 0.242·17-s + 0.580·18-s − 1.90·19-s + 1.41·20-s + 0.672·21-s + 0.507·22-s − 0.248·23-s − 1.03·24-s − 0.513·25-s − 0.468·26-s − 0.192·27-s − 2.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 5 | \( 1 - 1.55T + 5T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 - 0.965T + 11T^{2} \) |
| 13 | \( 1 + 0.969T + 13T^{2} \) |
| 19 | \( 1 + 8.30T + 19T^{2} \) |
| 23 | \( 1 + 1.19T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 + 3.46T + 37T^{2} \) |
| 41 | \( 1 + 8.18T + 41T^{2} \) |
| 43 | \( 1 - 5.09T + 43T^{2} \) |
| 47 | \( 1 - 0.481T + 47T^{2} \) |
| 53 | \( 1 - 4.00T + 53T^{2} \) |
| 59 | \( 1 + 6.66T + 59T^{2} \) |
| 61 | \( 1 + 6.07T + 61T^{2} \) |
| 67 | \( 1 + 9.49T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 7.67T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 0.199T + 83T^{2} \) |
| 89 | \( 1 - 8.81T + 89T^{2} \) |
| 97 | \( 1 - 9.96T + 97T^{2} \) |
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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
−6.88888708605793668977217633891, −6.51631343673092928262259442120, −6.00197810503311751465931884119, −5.50535988310226826055575035859, −4.64083909661931184989801415885, −4.02755105179402767014693480913, −3.32098190183170014568244648789, −2.44199548842069013179168571835, −1.72641160512631231894793871894, 0,
1.72641160512631231894793871894, 2.44199548842069013179168571835, 3.32098190183170014568244648789, 4.02755105179402767014693480913, 4.64083909661931184989801415885, 5.50535988310226826055575035859, 6.00197810503311751465931884119, 6.51631343673092928262259442120, 6.88888708605793668977217633891
