| L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s − 4.41·5-s + 2.41·6-s − 0.828·7-s − 4.41·8-s + 9-s + 10.6·10-s + 0.828·11-s − 3.82·12-s + 6·13-s + 1.99·14-s + 4.41·15-s + 2.99·16-s + 5.65·17-s − 2.41·18-s − 6·19-s − 16.8·20-s + 0.828·21-s − 1.99·22-s − 1.17·23-s + 4.41·24-s + 14.4·25-s − 14.4·26-s − 27-s − 3.17·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s − 1.97·5-s + 0.985·6-s − 0.313·7-s − 1.56·8-s + 0.333·9-s + 3.36·10-s + 0.249·11-s − 1.10·12-s + 1.66·13-s + 0.534·14-s + 1.13·15-s + 0.749·16-s + 1.37·17-s − 0.569·18-s − 1.37·19-s − 3.77·20-s + 0.180·21-s − 0.426·22-s − 0.244·23-s + 0.901·24-s + 2.89·25-s − 2.84·26-s − 0.192·27-s − 0.599·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 4.41T + 5T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 - 3.65T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 5.17T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 6.82T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 + 9.48T + 79T^{2} \) |
| 83 | \( 1 + 8.89T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 + 5.17T + 97T^{2} \) |
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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
−10.26101925336469903482717157532, −8.919574049056499123740364258834, −8.435323563057599259836859108174, −7.64606888432396291478797740471, −6.93662546972391500674019140369, −5.98251841523934946770834864207, −4.24662966285771311545591829551, −3.33909708714776672966603025857, −1.24627254262589795169697797038, 0,
1.24627254262589795169697797038, 3.33909708714776672966603025857, 4.24662966285771311545591829551, 5.98251841523934946770834864207, 6.93662546972391500674019140369, 7.64606888432396291478797740471, 8.435323563057599259836859108174, 8.919574049056499123740364258834, 10.26101925336469903482717157532
