| L(s) = 1 | − 2.35·2-s − 3.09·3-s + 3.56·4-s − 4.18·5-s + 7.29·6-s − 2.82·7-s − 3.69·8-s + 6.54·9-s + 9.86·10-s + 2.48·11-s − 11.0·12-s + 4.60·13-s + 6.67·14-s + 12.9·15-s + 1.59·16-s − 5.70·17-s − 15.4·18-s + 0.542·19-s − 14.9·20-s + 8.74·21-s − 5.86·22-s + 6.30·23-s + 11.4·24-s + 12.4·25-s − 10.8·26-s − 10.9·27-s − 10.0·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s − 1.78·3-s + 1.78·4-s − 1.87·5-s + 2.97·6-s − 1.06·7-s − 1.30·8-s + 2.18·9-s + 3.12·10-s + 0.749·11-s − 3.18·12-s + 1.27·13-s + 1.78·14-s + 3.33·15-s + 0.397·16-s − 1.38·17-s − 3.64·18-s + 0.124·19-s − 3.33·20-s + 1.90·21-s − 1.25·22-s + 1.31·23-s + 2.33·24-s + 2.49·25-s − 2.12·26-s − 2.11·27-s − 1.90·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 547 | \( 1 + T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 + 3.09T + 3T^{2} \) |
| 5 | \( 1 + 4.18T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 + 5.70T + 17T^{2} \) |
| 19 | \( 1 - 0.542T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 - 5.03T + 31T^{2} \) |
| 37 | \( 1 + 4.02T + 37T^{2} \) |
| 41 | \( 1 + 2.04T + 41T^{2} \) |
| 43 | \( 1 - 7.11T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 + 8.60T + 53T^{2} \) |
| 59 | \( 1 - 2.15T + 59T^{2} \) |
| 61 | \( 1 + 8.53T + 61T^{2} \) |
| 67 | \( 1 + 3.63T + 67T^{2} \) |
| 71 | \( 1 - 7.43T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 4.82T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 2.44T + 89T^{2} \) |
| 97 | \( 1 - 5.35T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.76166949042289388187837409603, −9.434270636438326087606800012568, −8.706779270795272429277612404325, −7.60852842286829346138178098350, −6.66860478662728389063772173467, −6.45872582097565305774566668301, −4.66547788379470256907594808681, −3.56994162980590643847882165835, −1.00546465244164347558091991985, 0,
1.00546465244164347558091991985, 3.56994162980590643847882165835, 4.66547788379470256907594808681, 6.45872582097565305774566668301, 6.66860478662728389063772173467, 7.60852842286829346138178098350, 8.706779270795272429277612404325, 9.434270636438326087606800012568, 10.76166949042289388187837409603
