L(s) = 1 | + (0.254 + 0.967i)2-s + (0.809 − 0.587i)3-s + (−0.870 + 0.491i)4-s + (0.985 − 0.170i)5-s + (0.774 + 0.633i)6-s + (−0.696 − 0.717i)8-s + (0.309 − 0.951i)9-s + (0.415 + 0.909i)10-s + (−0.415 + 0.909i)12-s + (−0.736 − 0.676i)13-s + (0.696 − 0.717i)15-s + (0.516 − 0.856i)16-s + (−0.0285 − 0.999i)17-s + (0.998 + 0.0570i)18-s + (0.610 − 0.791i)19-s + (−0.774 + 0.633i)20-s + ⋯ |
L(s) = 1 | + (0.254 + 0.967i)2-s + (0.809 − 0.587i)3-s + (−0.870 + 0.491i)4-s + (0.985 − 0.170i)5-s + (0.774 + 0.633i)6-s + (−0.696 − 0.717i)8-s + (0.309 − 0.951i)9-s + (0.415 + 0.909i)10-s + (−0.415 + 0.909i)12-s + (−0.736 − 0.676i)13-s + (0.696 − 0.717i)15-s + (0.516 − 0.856i)16-s + (−0.0285 − 0.999i)17-s + (0.998 + 0.0570i)18-s + (0.610 − 0.791i)19-s + (−0.774 + 0.633i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.107732024 - 0.4071780808i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.107732024 - 0.4071780808i\) |
\(L(1)\) |
\(\approx\) |
\(1.550439190 + 0.1302422410i\) |
\(L(1)\) |
\(\approx\) |
\(1.550439190 + 0.1302422410i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.254 + 0.967i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.985 - 0.170i)T \) |
| 13 | \( 1 + (-0.736 - 0.676i)T \) |
| 17 | \( 1 + (-0.0285 - 0.999i)T \) |
| 19 | \( 1 + (0.610 - 0.791i)T \) |
| 23 | \( 1 + (-0.654 - 0.755i)T \) |
| 29 | \( 1 + (-0.941 - 0.336i)T \) |
| 31 | \( 1 + (-0.993 + 0.113i)T \) |
| 37 | \( 1 + (0.198 - 0.980i)T \) |
| 41 | \( 1 + (-0.362 + 0.931i)T \) |
| 43 | \( 1 + (0.142 + 0.989i)T \) |
| 47 | \( 1 + (0.998 - 0.0570i)T \) |
| 53 | \( 1 + (0.516 + 0.856i)T \) |
| 59 | \( 1 + (0.362 + 0.931i)T \) |
| 61 | \( 1 + (-0.254 + 0.967i)T \) |
| 67 | \( 1 + (0.841 - 0.540i)T \) |
| 71 | \( 1 + (-0.564 - 0.825i)T \) |
| 73 | \( 1 + (0.0855 + 0.996i)T \) |
| 79 | \( 1 + (-0.897 - 0.441i)T \) |
| 83 | \( 1 + (-0.921 - 0.389i)T \) |
| 89 | \( 1 + (0.959 + 0.281i)T \) |
| 97 | \( 1 + (0.985 + 0.170i)T \) |
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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
−21.99011568651644207589936720026, −21.40563553521568525345679953062, −20.540707135773765639171017103926, −20.06784748767918391040662264987, −19.00583861059717883301381431676, −18.57036308842574932923049285284, −17.3970754629108820956714253513, −16.720146092728053300084236008772, −15.43536377802675816432911413888, −14.50945003669779575731781932849, −14.14004701483336539659554983491, −13.310013814162247980079009475, −12.540527238745277515319711685890, −11.4481946874417035215889165891, −10.46077090123941987031189812052, −9.89182111116389100723291557594, −9.2773552534935722985371553053, −8.4506716433651272931356943996, −7.2566343544512349482940335116, −5.80967740495857918914305597468, −5.14338198490186626825911835541, −3.98803971109269370315399239937, −3.33346748978356449832517521144, −2.08859247323549563894299321531, −1.74252456389807675303288472253,
0.78445571172673937876734419788, 2.31169640750009246691129178164, 3.069844110924640008684424980931, 4.375632625189131758857982426934, 5.37369831038000183142806277603, 6.154000790096128373047852865489, 7.204984877544587233241754381273, 7.6572382566737987119750243817, 8.871140960290838257503836246663, 9.33166428858643053367198374668, 10.17865728170577435835191193189, 11.81251588014442580280551369801, 12.823788491920411936678576846290, 13.23202003585113762860390807070, 14.129227600842350110507660658457, 14.587342337285065191856751711814, 15.49992579828748673785958323529, 16.44491593919701429167714663256, 17.304249871467887690325490288377, 18.20332718325086383817167241603, 18.35437773026879796658687799948, 19.78493084997824437185363159578, 20.4385125282976054675125004529, 21.38206606481420857287687110645, 22.15389213648594823770310590383