L(s) = 1 | + (0.935 + 0.352i)3-s + (−0.683 − 0.729i)5-s + (0.751 + 0.659i)9-s + (0.812 − 0.582i)11-s + (0.956 − 0.290i)13-s + (−0.382 − 0.923i)15-s + (0.991 − 0.130i)17-s + (0.528 + 0.849i)19-s + (−0.997 + 0.0654i)23-s + (−0.0654 + 0.997i)25-s + (0.471 + 0.881i)27-s + (−0.995 − 0.0980i)29-s + (0.965 + 0.258i)31-s + (0.965 − 0.258i)33-s + (0.0327 − 0.999i)37-s + ⋯ |
L(s) = 1 | + (0.935 + 0.352i)3-s + (−0.683 − 0.729i)5-s + (0.751 + 0.659i)9-s + (0.812 − 0.582i)11-s + (0.956 − 0.290i)13-s + (−0.382 − 0.923i)15-s + (0.991 − 0.130i)17-s + (0.528 + 0.849i)19-s + (−0.997 + 0.0654i)23-s + (−0.0654 + 0.997i)25-s + (0.471 + 0.881i)27-s + (−0.995 − 0.0980i)29-s + (0.965 + 0.258i)31-s + (0.965 − 0.258i)33-s + (0.0327 − 0.999i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.377758011 - 0.1045923081i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.377758011 - 0.1045923081i\) |
\(L(1)\) |
\(\approx\) |
\(1.499449011 + 0.01645857547i\) |
\(L(1)\) |
\(\approx\) |
\(1.499449011 + 0.01645857547i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.935 + 0.352i)T \) |
| 5 | \( 1 + (-0.683 - 0.729i)T \) |
| 11 | \( 1 + (0.812 - 0.582i)T \) |
| 13 | \( 1 + (0.956 - 0.290i)T \) |
| 17 | \( 1 + (0.991 - 0.130i)T \) |
| 19 | \( 1 + (0.528 + 0.849i)T \) |
| 23 | \( 1 + (-0.997 + 0.0654i)T \) |
| 29 | \( 1 + (-0.995 - 0.0980i)T \) |
| 31 | \( 1 + (0.965 + 0.258i)T \) |
| 37 | \( 1 + (0.0327 - 0.999i)T \) |
| 41 | \( 1 + (-0.831 + 0.555i)T \) |
| 43 | \( 1 + (0.773 + 0.634i)T \) |
| 47 | \( 1 + (-0.793 + 0.608i)T \) |
| 53 | \( 1 + (0.582 + 0.812i)T \) |
| 59 | \( 1 + (-0.227 - 0.973i)T \) |
| 61 | \( 1 + (0.986 - 0.162i)T \) |
| 67 | \( 1 + (0.352 - 0.935i)T \) |
| 71 | \( 1 + (-0.195 - 0.980i)T \) |
| 73 | \( 1 + (-0.321 - 0.946i)T \) |
| 79 | \( 1 + (0.130 - 0.991i)T \) |
| 83 | \( 1 + (-0.881 - 0.471i)T \) |
| 89 | \( 1 + (0.442 + 0.896i)T \) |
| 97 | \( 1 + (0.707 - 0.707i)T \) |
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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
−20.195633281730917762644190339627, −19.42154932660278056749123454997, −18.8271484893532556485940817457, −18.27104059916474394066055628166, −17.45320347857781314445355869975, −16.36683124225054917117029593465, −15.55767117741512019392903714484, −15.020971728707904470242186508819, −14.25885047654800555804681388082, −13.742337953460632726577549742190, −12.83663942476487217601173051095, −11.84268855653725949450594871863, −11.52058152608207341787815774030, −10.22880202686806100193631821749, −9.69967797437498399407591911261, −8.64431719694587825284337680585, −8.12181347178996781718424888030, −7.12015238387571789664890972476, −6.79764438048188908787993792, −5.71209647312721438773402592348, −4.25806308442351023594255366736, −3.761297546687300258815755487842, −2.97894614394439880170428531854, −1.999869508385597562843862711491, −1.02172794042716246610182552000,
0.99510170615167932953618400199, 1.77278818269968662797088629912, 3.28372858403276435337765857094, 3.61942466204768192317463782193, 4.42535972361276849035108526373, 5.45816231141065331451569111425, 6.31238312445842888807458732266, 7.73832093665386744929973393689, 7.93206334005136867898899808974, 8.83404595362312523078758218058, 9.4426568205344638800562695711, 10.278977088794151861400515752658, 11.26657629984519234082696978393, 12.00977296601613717052742669436, 12.79084766330178891777449000990, 13.652430433166153941652265181505, 14.25860603422216430910205612125, 14.98478911357225346538758285941, 16.00098476802133415229102185516, 16.18960489516246451612542124484, 17.0098063681661582284524441693, 18.209481134704262454803195306721, 18.942835360706328280763004053287, 19.52552478200477588178631845090, 20.25322749792297334652177621025