| L(s) = 1 | − 2-s + 4-s − 5-s − 5·7-s − 8-s + 10-s + 11-s − 4·13-s + 5·14-s + 16-s − 4·19-s − 20-s − 22-s − 3·23-s + 25-s + 4·26-s − 5·28-s + 9·29-s − 5·31-s − 32-s + 5·35-s − 2·37-s + 4·38-s + 40-s + 11·43-s + 44-s + 3·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.88·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 1.10·13-s + 1.33·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s − 0.213·22-s − 0.625·23-s + 1/5·25-s + 0.784·26-s − 0.944·28-s + 1.67·29-s − 0.898·31-s − 0.176·32-s + 0.845·35-s − 0.328·37-s + 0.648·38-s + 0.158·40-s + 1.67·43-s + 0.150·44-s + 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.75586275903351, −12.52486284790728, −12.08756821956467, −11.66241561353716, −11.00557866749278, −10.49746541623274, −10.12402790982669, −9.770103803484036, −9.321098397009920, −8.809276938422186, −8.507011108514295, −7.779350485323045, −7.227397445520979, −7.072048686471763, −6.318287703703526, −6.164136847961689, −5.576484727863437, −4.692041898278294, −4.303986737668335, −3.643671091145026, −3.158528883247662, −2.621091570042912, −2.205291578473418, −1.298020080008979, −0.4756111545892521, 0,
0.4756111545892521, 1.298020080008979, 2.205291578473418, 2.621091570042912, 3.158528883247662, 3.643671091145026, 4.303986737668335, 4.692041898278294, 5.576484727863437, 6.164136847961689, 6.318287703703526, 7.072048686471763, 7.227397445520979, 7.779350485323045, 8.507011108514295, 8.809276938422186, 9.321098397009920, 9.770103803484036, 10.12402790982669, 10.49746541623274, 11.00557866749278, 11.66241561353716, 12.08756821956467, 12.52486284790728, 12.75586275903351
