L(s) = 1 | − 2.81·3-s + 0.0468·7-s + 4.91·9-s + 0.387·11-s + 6.45·13-s + 2.04·17-s + 6.02·19-s − 0.131·21-s − 6.77·23-s − 5.39·27-s − 4.96·29-s − 3.11·31-s − 1.08·33-s − 10.2·37-s − 18.1·39-s + 10.5·41-s − 3.83·43-s − 4.40·47-s − 6.99·49-s − 5.74·51-s + 7.46·53-s − 16.9·57-s − 5.89·59-s + 0.905·61-s + 0.230·63-s − 15.2·67-s + 19.0·69-s + ⋯ |
L(s) = 1 | − 1.62·3-s + 0.0177·7-s + 1.63·9-s + 0.116·11-s + 1.78·13-s + 0.495·17-s + 1.38·19-s − 0.0287·21-s − 1.41·23-s − 1.03·27-s − 0.921·29-s − 0.559·31-s − 0.189·33-s − 1.68·37-s − 2.90·39-s + 1.64·41-s − 0.584·43-s − 0.642·47-s − 0.999·49-s − 0.804·51-s + 1.02·53-s − 2.24·57-s − 0.767·59-s + 0.115·61-s + 0.0290·63-s − 1.86·67-s + 2.29·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 3 | \( 1 + 2.81T + 3T^{2} \) |
| 7 | \( 1 - 0.0468T + 7T^{2} \) |
| 11 | \( 1 - 0.387T + 11T^{2} \) |
| 13 | \( 1 - 6.45T + 13T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 19 | \( 1 - 6.02T + 19T^{2} \) |
| 23 | \( 1 + 6.77T + 23T^{2} \) |
| 29 | \( 1 + 4.96T + 29T^{2} \) |
| 31 | \( 1 + 3.11T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 3.83T + 43T^{2} \) |
| 47 | \( 1 + 4.40T + 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 + 5.89T + 59T^{2} \) |
| 61 | \( 1 - 0.905T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 9.61T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 1.87T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 - 5.66T + 89T^{2} \) |
| 97 | \( 1 - 9.99T + 97T^{2} \) |
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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
−7.22065671628754816882767527892, −6.40628531278077454748069415777, −5.82626108604197970646827154617, −5.56196426564142895050983239573, −4.69908631896745178918507672338, −3.85293015578830689373763211097, −3.29477561183764039883970466437, −1.73489238375750046925294680798, −1.12178895845873542870869167081, 0,
1.12178895845873542870869167081, 1.73489238375750046925294680798, 3.29477561183764039883970466437, 3.85293015578830689373763211097, 4.69908631896745178918507672338, 5.56196426564142895050983239573, 5.82626108604197970646827154617, 6.40628531278077454748069415777, 7.22065671628754816882767527892