L(s) = 1 | − 2-s + (0.973 − 0.230i)3-s + 4-s + (0.0581 + 0.998i)5-s + (−0.973 + 0.230i)6-s + (0.893 + 0.448i)7-s − 8-s + (0.893 − 0.448i)9-s + (−0.0581 − 0.998i)10-s + (−0.0581 + 0.998i)11-s + (0.973 − 0.230i)12-s + (0.0581 + 0.998i)13-s + (−0.893 − 0.448i)14-s + (0.286 + 0.957i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | − 2-s + (0.973 − 0.230i)3-s + 4-s + (0.0581 + 0.998i)5-s + (−0.973 + 0.230i)6-s + (0.893 + 0.448i)7-s − 8-s + (0.893 − 0.448i)9-s + (−0.0581 − 0.998i)10-s + (−0.0581 + 0.998i)11-s + (0.973 − 0.230i)12-s + (0.0581 + 0.998i)13-s + (−0.893 − 0.448i)14-s + (0.286 + 0.957i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.606310248 + 1.142802190i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.606310248 + 1.142802190i\) |
\(L(1)\) |
\(\approx\) |
\(1.118848708 + 0.2885881784i\) |
\(L(1)\) |
\(\approx\) |
\(1.118848708 + 0.2885881784i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.973 - 0.230i)T \) |
| 5 | \( 1 + (0.0581 + 0.998i)T \) |
| 7 | \( 1 + (0.893 + 0.448i)T \) |
| 11 | \( 1 + (-0.0581 + 0.998i)T \) |
| 13 | \( 1 + (0.0581 + 0.998i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.766 - 0.642i)T \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.893 - 0.448i)T \) |
| 31 | \( 1 + (0.993 - 0.116i)T \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.766 + 0.642i)T \) |
| 47 | \( 1 + (0.973 + 0.230i)T \) |
| 53 | \( 1 + (-0.686 - 0.727i)T \) |
| 59 | \( 1 + (0.286 + 0.957i)T \) |
| 61 | \( 1 + (-0.597 + 0.802i)T \) |
| 67 | \( 1 + (0.973 + 0.230i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.893 - 0.448i)T \) |
| 79 | \( 1 + (0.286 + 0.957i)T \) |
| 83 | \( 1 + (0.396 + 0.918i)T \) |
| 89 | \( 1 + (0.835 - 0.549i)T \) |
| 97 | \( 1 + (0.993 + 0.116i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.65356203125347286197437080413, −17.42012524407816549101037239978, −17.03738285654279850163824901243, −16.50898837095604767451141088996, −15.55811663301892145704873342724, −15.15727862440588985391667702704, −14.37934660836528476280133642442, −13.57131295747755034861070361674, −12.85994905815728316245029253662, −12.19598389290097628147920402729, −11.1563288254085621690672589509, −10.57487408078839226938876030743, −10.004605986427488774056180355777, −9.098400139966688443780819463477, −8.507846391423318362978877475047, −8.06248113577558455337862562181, −7.68984738729851731569784212788, −6.54516895306687396197636068172, −5.56369218555223392936525845915, −4.912505425234797656273749675992, −3.73854713250344527241001907524, −3.28131635861701467635155129796, −2.07497215845226258204824935162, −1.45889287802264362593380629245, −0.71637167759080075815484758849,
1.11121040881663983099305735276, 2.187593717127567771450481565536, 2.25791362685440647299011829815, 3.19412912152065551676986539290, 4.21662722558064129344901999271, 5.134544385733354966843792732427, 6.446532616467152573455961585467, 6.88841428048878242516328904519, 7.52255608374744538001789387312, 8.1208763406936666555310015828, 8.96120603341361910039343193437, 9.46621457253908440561078324156, 10.14022182767062698320730004622, 10.94736608613898217783467464194, 11.638635384094175557787920421428, 12.18812481025740181999401745900, 13.217606734568085524879406820310, 14.11725312894142763358205545968, 14.66470577905691094975336357915, 15.25812856790783262347507592225, 15.56596800700732594249276315811, 16.7934753515139444071802705228, 17.41734207150424218147921055731, 18.209297498599689904268036661378, 18.56761784693215373603415318649