L(s) = 1 | − 2·3-s + 5-s + 3.46·7-s + 9-s − 2·15-s + 6.92·17-s − 6.92·19-s − 6.92·21-s − 6·23-s + 25-s + 4·27-s − 4·31-s + 3.46·35-s + 10·37-s − 6.92·41-s − 3.46·43-s + 45-s + 6·47-s + 4.99·49-s − 13.8·51-s − 6·53-s + 13.8·57-s − 6.92·61-s + 3.46·63-s − 10·67-s + 12·69-s + 6.92·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1.30·7-s + 0.333·9-s − 0.516·15-s + 1.68·17-s − 1.58·19-s − 1.51·21-s − 1.25·23-s + 0.200·25-s + 0.769·27-s − 0.718·31-s + 0.585·35-s + 1.64·37-s − 1.08·41-s − 0.528·43-s + 0.149·45-s + 0.875·47-s + 0.714·49-s − 1.94·51-s − 0.824·53-s + 1.83·57-s − 0.887·61-s + 0.436·63-s − 1.22·67-s + 1.44·69-s + 0.810·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.30257145429964989470127056321, −6.46481495347833655471505935935, −5.77020538277290503662835174098, −5.52003923703056519246466974515, −4.63751013167091288395795264502, −4.14712762057067364199079420065, −2.95283724871260692501086256254, −1.90207122957652895029283014155, −1.24669399941346977336697117269, 0,
1.24669399941346977336697117269, 1.90207122957652895029283014155, 2.95283724871260692501086256254, 4.14712762057067364199079420065, 4.63751013167091288395795264502, 5.52003923703056519246466974515, 5.77020538277290503662835174098, 6.46481495347833655471505935935, 7.30257145429964989470127056321