| L(s) = 1 | + 2-s + 2.71·3-s + 4-s − 3.56·5-s + 2.71·6-s + 8-s + 4.36·9-s − 3.56·10-s − 3.83·11-s + 2.71·12-s + 1.23·13-s − 9.66·15-s + 16-s − 7.36·17-s + 4.36·18-s − 1.14·19-s − 3.56·20-s − 3.83·22-s − 4.15·23-s + 2.71·24-s + 7.67·25-s + 1.23·26-s + 3.70·27-s − 4.30·29-s − 9.66·30-s − 3.42·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.56·3-s + 0.5·4-s − 1.59·5-s + 1.10·6-s + 0.353·8-s + 1.45·9-s − 1.12·10-s − 1.15·11-s + 0.783·12-s + 0.342·13-s − 2.49·15-s + 0.250·16-s − 1.78·17-s + 1.02·18-s − 0.262·19-s − 0.796·20-s − 0.816·22-s − 0.865·23-s + 0.553·24-s + 1.53·25-s + 0.242·26-s + 0.712·27-s − 0.798·29-s − 1.76·30-s − 0.615·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 11 | \( 1 + 3.83T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 7.36T + 17T^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 + 4.15T + 23T^{2} \) |
| 29 | \( 1 + 4.30T + 29T^{2} \) |
| 31 | \( 1 + 3.42T + 31T^{2} \) |
| 37 | \( 1 - 0.0352T + 37T^{2} \) |
| 43 | \( 1 - 3.45T + 43T^{2} \) |
| 47 | \( 1 - 0.688T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 7.18T + 67T^{2} \) |
| 71 | \( 1 - 5.49T + 71T^{2} \) |
| 73 | \( 1 + 7.25T + 73T^{2} \) |
| 79 | \( 1 + 6.96T + 79T^{2} \) |
| 83 | \( 1 - 9.63T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−7.88518754706409050217236033771, −7.61208323665264813725694444678, −6.89184473175904778456618824368, −5.80777117120269847113286660052, −4.59051621854345363365230600706, −4.15036202566815725245779021247, −3.47328900291819184960850450607, −2.70433436919293725093270632579, −1.95330744311661517560650017350, 0,
1.95330744311661517560650017350, 2.70433436919293725093270632579, 3.47328900291819184960850450607, 4.15036202566815725245779021247, 4.59051621854345363365230600706, 5.80777117120269847113286660052, 6.89184473175904778456618824368, 7.61208323665264813725694444678, 7.88518754706409050217236033771
