L(s) = 1 | + (0.483 − 1.32i)3-s + (0.173 + 0.984i)5-s + (−0.5 + 0.866i)7-s + (−0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (1.39 + 0.245i)15-s + (0.766 − 0.642i)17-s + (0.909 + 1.08i)21-s + (0.909 − 1.08i)29-s + (1.39 − 0.245i)33-s + (−0.939 − 0.342i)35-s + 1.41i·37-s + (−0.483 + 1.32i)41-s + (−0.173 − 0.984i)43-s + (0.500 − 0.866i)45-s + ⋯ |
L(s) = 1 | + (0.483 − 1.32i)3-s + (0.173 + 0.984i)5-s + (−0.5 + 0.866i)7-s + (−0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (1.39 + 0.245i)15-s + (0.766 − 0.642i)17-s + (0.909 + 1.08i)21-s + (0.909 − 1.08i)29-s + (1.39 − 0.245i)33-s + (−0.939 − 0.342i)35-s + 1.41i·37-s + (−0.483 + 1.32i)41-s + (−0.173 − 0.984i)43-s + (0.500 − 0.866i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.310357577\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.310357577\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.483 + 1.32i)T + (-0.766 - 0.642i)T^{2} \) |
| 5 | \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.909 + 1.08i)T + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 + (0.483 - 1.32i)T + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (0.909 + 1.08i)T + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (1.39 - 0.245i)T + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
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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
−9.753118314788861650360692655855, −8.758029292418522682948167989881, −7.971801917069062244808183618562, −7.15686313696971675203270054552, −6.60335177267628717211780271924, −5.99283416292580919612046404787, −4.69553472719243925664247327114, −3.13323400073370687336146945923, −2.62287178525519226845103481225, −1.56373271096809404512043635080,
1.22116492764177397244299601523, 3.13352908739783390120440584303, 3.76670139338667314526271631750, 4.55083797475651139481127504856, 5.38677261224167027607381449148, 6.34307560320525915669872499015, 7.47804527452552559164329625091, 8.552954720792161061556540808477, 8.953839568047352857084528029676, 9.682495364040144313230910102622