| L(s) = 1 | − 2·2-s − 4·3-s + 4·4-s − 5·5-s + 8·6-s − 8·8-s − 11·9-s + 10·10-s + 60·11-s − 16·12-s − 38·13-s + 20·15-s + 16·16-s − 42·17-s + 22·18-s + 52·19-s − 20·20-s − 120·22-s + 120·23-s + 32·24-s + 25·25-s + 76·26-s + 152·27-s − 234·29-s − 40·30-s + 304·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.769·3-s + 1/2·4-s − 0.447·5-s + 0.544·6-s − 0.353·8-s − 0.407·9-s + 0.316·10-s + 1.64·11-s − 0.384·12-s − 0.810·13-s + 0.344·15-s + 1/4·16-s − 0.599·17-s + 0.288·18-s + 0.627·19-s − 0.223·20-s − 1.16·22-s + 1.08·23-s + 0.272·24-s + 1/5·25-s + 0.573·26-s + 1.08·27-s − 1.49·29-s − 0.243·30-s + 1.76·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 234 T + p^{3} T^{2} \) |
| 31 | \( 1 - 304 T + p^{3} T^{2} \) |
| 37 | \( 1 + 106 T + p^{3} T^{2} \) |
| 41 | \( 1 - 54 T + p^{3} T^{2} \) |
| 43 | \( 1 + 196 T + p^{3} T^{2} \) |
| 47 | \( 1 + 336 T + p^{3} T^{2} \) |
| 53 | \( 1 - 438 T + p^{3} T^{2} \) |
| 59 | \( 1 - 444 T + p^{3} T^{2} \) |
| 61 | \( 1 + 38 T + p^{3} T^{2} \) |
| 67 | \( 1 + 988 T + p^{3} T^{2} \) |
| 71 | \( 1 + 720 T + p^{3} T^{2} \) |
| 73 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 79 | \( 1 + 808 T + p^{3} T^{2} \) |
| 83 | \( 1 + 612 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1146 T + p^{3} T^{2} \) |
| 97 | \( 1 - 70 T + p^{3} T^{2} \) |
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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
−10.08166994191718082599212834472, −9.185738291158917633618622049403, −8.466737067958872578142520875842, −7.20212487840482341692703094069, −6.61046836033739617668425968860, −5.51176278702901317192354142325, −4.34036861271575492935150930333, −2.95885373404145523362266096864, −1.27809098940068457354395916273, 0,
1.27809098940068457354395916273, 2.95885373404145523362266096864, 4.34036861271575492935150930333, 5.51176278702901317192354142325, 6.61046836033739617668425968860, 7.20212487840482341692703094069, 8.466737067958872578142520875842, 9.185738291158917633618622049403, 10.08166994191718082599212834472
