L(s) = 1 | + 2.71·3-s − 0.592·5-s − 2.94·7-s + 4.39·9-s + 2.40·11-s + 3.66·13-s − 1.61·15-s − 7.66·17-s + 19-s − 8.01·21-s − 7.91·23-s − 4.64·25-s + 3.80·27-s − 8.53·29-s − 0.177·31-s + 6.54·33-s + 1.74·35-s + 4.75·37-s + 9.97·39-s + 4.20·41-s − 5.39·43-s − 2.60·45-s + 0.161·47-s + 1.67·49-s − 20.8·51-s − 11.3·53-s − 1.42·55-s + ⋯ |
L(s) = 1 | + 1.57·3-s − 0.264·5-s − 1.11·7-s + 1.46·9-s + 0.725·11-s + 1.01·13-s − 0.415·15-s − 1.85·17-s + 0.229·19-s − 1.74·21-s − 1.65·23-s − 0.929·25-s + 0.731·27-s − 1.58·29-s − 0.0318·31-s + 1.13·33-s + 0.294·35-s + 0.782·37-s + 1.59·39-s + 0.656·41-s − 0.823·43-s − 0.388·45-s + 0.0235·47-s + 0.239·49-s − 2.91·51-s − 1.55·53-s − 0.192·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 0.592T + 5T^{2} \) |
| 7 | \( 1 + 2.94T + 7T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 13 | \( 1 - 3.66T + 13T^{2} \) |
| 17 | \( 1 + 7.66T + 17T^{2} \) |
| 23 | \( 1 + 7.91T + 23T^{2} \) |
| 29 | \( 1 + 8.53T + 29T^{2} \) |
| 31 | \( 1 + 0.177T + 31T^{2} \) |
| 37 | \( 1 - 4.75T + 37T^{2} \) |
| 41 | \( 1 - 4.20T + 41T^{2} \) |
| 43 | \( 1 + 5.39T + 43T^{2} \) |
| 47 | \( 1 - 0.161T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 3.61T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 9.67T + 71T^{2} \) |
| 73 | \( 1 - 6.62T + 73T^{2} \) |
| 83 | \( 1 + 4.26T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.940299046130115591197955654326, −7.08953608094537080400961407560, −6.41613693157465398983821242258, −5.83883779343853370444574866679, −4.32200082465518569016628967770, −3.86501298520073142067378894130, −3.36267234756899549619687684440, −2.36305121162220710975649769525, −1.69022293527134900728226201005, 0,
1.69022293527134900728226201005, 2.36305121162220710975649769525, 3.36267234756899549619687684440, 3.86501298520073142067378894130, 4.32200082465518569016628967770, 5.83883779343853370444574866679, 6.41613693157465398983821242258, 7.08953608094537080400961407560, 7.940299046130115591197955654326