| L(s) = 1 | + 2-s − 2·3-s + 4-s + 2·5-s − 2·6-s + 7-s + 8-s + 9-s + 2·10-s − 2·11-s − 2·12-s − 4·13-s + 14-s − 4·15-s + 16-s + 6·17-s + 18-s + 2·20-s − 2·21-s − 2·22-s + 8·23-s − 2·24-s − 25-s − 4·26-s + 4·27-s + 28-s + 4·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.603·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.447·20-s − 0.436·21-s − 0.426·22-s + 1.66·23-s − 0.408·24-s − 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.188·28-s + 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.29898152727639, −12.67730952344874, −12.17369340683834, −12.02767666357644, −11.59028520604953, −10.86635314433652, −10.52684971860444, −10.14172839510613, −9.809631004904652, −9.064711299860372, −8.507157050997991, −7.895799333306955, −7.341636432338946, −6.878428809408180, −6.457187535259841, −5.735867456205933, −5.509233211368465, −5.048350814985304, −4.808019833964352, −4.088391282323243, −3.196588018346276, −2.759738189263197, −2.289361376134564, −1.354658014142735, −0.9706392782284377, 0,
0.9706392782284377, 1.354658014142735, 2.289361376134564, 2.759738189263197, 3.196588018346276, 4.088391282323243, 4.808019833964352, 5.048350814985304, 5.509233211368465, 5.735867456205933, 6.457187535259841, 6.878428809408180, 7.341636432338946, 7.895799333306955, 8.507157050997991, 9.064711299860372, 9.809631004904652, 10.14172839510613, 10.52684971860444, 10.86635314433652, 11.59028520604953, 12.02767666357644, 12.17369340683834, 12.67730952344874, 13.29898152727639
