| L(s) = 1 | − 2-s − 2·3-s + 4-s + 5-s + 2·6-s − 8-s + 9-s − 10-s − 11-s − 2·12-s − 13-s − 2·15-s + 16-s − 4·17-s − 18-s + 2·19-s + 20-s + 22-s + 6·23-s + 2·24-s + 25-s + 26-s + 4·27-s − 6·29-s + 2·30-s + 4·31-s − 32-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 − 0.301·11-s − 0.577·12-s − 0.277·13-s − 0.516·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.213·22-s + 1.25·23-s + 0.408·24-s + 1/5·25-s + 0.196·26-s + 0.769·27-s − 1.11·29-s + 0.365·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.49155369574655, −13.92247111437083, −13.14166171103266, −12.81040981019705, −12.40534199547065, −11.59974624007366, −11.21092369691116, −11.05356355650240, −10.37702414375447, −9.907052508738145, −9.325569941386450, −8.916679402230557, −8.318919969132204, −7.610594599380998, −7.107012242169404, −6.590433432972939, −6.141797418786464, −5.470518213666110, −5.135062750550537, −4.507124663599344, −3.684087810857504, −2.782292385980177, −2.362277793807341, −1.438426837894697, −0.7668702148354610, 0,
0.7668702148354610, 1.438426837894697, 2.362277793807341, 2.782292385980177, 3.684087810857504, 4.507124663599344, 5.135062750550537, 5.470518213666110, 6.141797418786464, 6.590433432972939, 7.107012242169404, 7.610594599380998, 8.318919969132204, 8.916679402230557, 9.325569941386450, 9.907052508738145, 10.37702414375447, 11.05356355650240, 11.21092369691116, 11.59974624007366, 12.40534199547065, 12.81040981019705, 13.14166171103266, 13.92247111437083, 14.49155369574655
