L(s) = 1 | − 5-s + 2.56·7-s − 2·11-s − 3.56·13-s − 2.56·17-s + 6·19-s − 23-s + 25-s − 6.12·29-s + 7.24·31-s − 2.56·35-s − 4.56·37-s − 4.12·41-s − 4.68·47-s − 0.438·49-s + 4.56·53-s + 2·55-s + 3.68·59-s − 7.12·61-s + 3.56·65-s − 8.56·67-s − 10.1·71-s − 4.43·73-s − 5.12·77-s + 4.87·79-s + 13.9·83-s + 2.56·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.968·7-s − 0.603·11-s − 0.987·13-s − 0.621·17-s + 1.37·19-s − 0.208·23-s + 0.200·25-s − 1.13·29-s + 1.30·31-s − 0.432·35-s − 0.749·37-s − 0.643·41-s − 0.683·47-s − 0.0626·49-s + 0.626·53-s + 0.269·55-s + 0.479·59-s − 0.912·61-s + 0.441·65-s − 1.04·67-s − 1.20·71-s − 0.519·73-s − 0.583·77-s + 0.548·79-s + 1.52·83-s + 0.277·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 29 | \( 1 + 6.12T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 - 4.56T + 53T^{2} \) |
| 59 | \( 1 - 3.68T + 59T^{2} \) |
| 61 | \( 1 + 7.12T + 61T^{2} \) |
| 67 | \( 1 + 8.56T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 - 4.87T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.921854725246817337956598359971, −7.49170424692968846539947185336, −6.77012186769738405457857169435, −5.63895149623702245207533746629, −4.98991005742718407600633623828, −4.43471684875828998944963263095, −3.35188385305982706312263806678, −2.46693695378323062328152800308, −1.44335279645306814673893110777, 0,
1.44335279645306814673893110777, 2.46693695378323062328152800308, 3.35188385305982706312263806678, 4.43471684875828998944963263095, 4.98991005742718407600633623828, 5.63895149623702245207533746629, 6.77012186769738405457857169435, 7.49170424692968846539947185336, 7.921854725246817337956598359971