| L(s) = 1 | − 1.35·2-s − 0.242·3-s − 0.159·4-s + 5-s + 0.329·6-s − 7-s + 2.92·8-s − 2.94·9-s − 1.35·10-s + 0.0386·12-s − 2.34·13-s + 1.35·14-s − 0.242·15-s − 3.65·16-s − 0.927·17-s + 3.99·18-s + 1.95·19-s − 0.159·20-s + 0.242·21-s + 2.68·23-s − 0.711·24-s + 25-s + 3.18·26-s + 1.44·27-s + 0.159·28-s + 0.245·29-s + 0.329·30-s + ⋯ |
| L(s) = 1 | − 0.959·2-s − 0.140·3-s − 0.0795·4-s + 0.447·5-s + 0.134·6-s − 0.377·7-s + 1.03·8-s − 0.980·9-s − 0.429·10-s + 0.0111·12-s − 0.650·13-s + 0.362·14-s − 0.0627·15-s − 0.914·16-s − 0.224·17-s + 0.940·18-s + 0.448·19-s − 0.0355·20-s + 0.0530·21-s + 0.559·23-s − 0.145·24-s + 0.200·25-s + 0.624·26-s + 0.277·27-s + 0.0300·28-s + 0.0456·29-s + 0.0601·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 3 | \( 1 + 0.242T + 3T^{2} \) |
| 13 | \( 1 + 2.34T + 13T^{2} \) |
| 17 | \( 1 + 0.927T + 17T^{2} \) |
| 19 | \( 1 - 1.95T + 19T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 - 0.245T + 29T^{2} \) |
| 31 | \( 1 - 2.99T + 31T^{2} \) |
| 37 | \( 1 - 6.23T + 37T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 + 9.64T + 43T^{2} \) |
| 47 | \( 1 - 5.84T + 47T^{2} \) |
| 53 | \( 1 + 4.58T + 53T^{2} \) |
| 59 | \( 1 + 9.66T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 + 2.21T + 71T^{2} \) |
| 73 | \( 1 + 0.100T + 73T^{2} \) |
| 79 | \( 1 + 6.31T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 2.73T + 89T^{2} \) |
| 97 | \( 1 - 3.11T + 97T^{2} \) |
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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
−8.194941677398240549675313750463, −7.45987668873723744552414132907, −6.69661782997850585368351299552, −5.87382993973365391984538623364, −5.10339136200565389193831863709, −4.36850659751532971765998604641, −3.14674602602346814070299842064, −2.34263378196145346687608772081, −1.10975260985523699320964583679, 0,
1.10975260985523699320964583679, 2.34263378196145346687608772081, 3.14674602602346814070299842064, 4.36850659751532971765998604641, 5.10339136200565389193831863709, 5.87382993973365391984538623364, 6.69661782997850585368351299552, 7.45987668873723744552414132907, 8.194941677398240549675313750463
