| L(s) = 1 | − 2-s + 2·3-s + 4-s + 5-s − 2·6-s − 8-s + 9-s − 10-s − 2·11-s + 2·12-s + 2·15-s + 16-s + 2·17-s − 18-s + 4·19-s + 20-s + 2·22-s − 4·23-s − 2·24-s + 25-s − 4·27-s − 4·29-s − 2·30-s + 31-s − 32-s − 4·33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.577·12-s + 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.426·22-s − 0.834·23-s − 0.408·24-s + 1/5·25-s − 0.769·27-s − 0.742·29-s − 0.365·30-s + 0.179·31-s − 0.176·32-s − 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.82175761839282, −14.18430043340641, −13.79995860836808, −13.27779306121418, −12.83323977283085, −12.11461479593929, −11.57760080023914, −11.09485327728227, −10.33896168484625, −9.899673827118826, −9.566707715804105, −9.000511349882592, −8.459707688912190, −8.009316321322461, −7.484353131788855, −7.159482514528456, −6.096210613629191, −5.855956468907525, −5.088162059013794, −4.327042745370921, −3.441299013832088, −3.089521829671435, −2.378339529624240, −1.867727773542795, −1.077022665857124, 0,
1.077022665857124, 1.867727773542795, 2.378339529624240, 3.089521829671435, 3.441299013832088, 4.327042745370921, 5.088162059013794, 5.855956468907525, 6.096210613629191, 7.159482514528456, 7.484353131788855, 8.009316321322461, 8.459707688912190, 9.000511349882592, 9.566707715804105, 9.899673827118826, 10.33896168484625, 11.09485327728227, 11.57760080023914, 12.11461479593929, 12.83323977283085, 13.27779306121418, 13.79995860836808, 14.18430043340641, 14.82175761839282
