| L(s) = 1 | + 0.618i·2-s − i·3-s + 1.61·4-s + 0.618·6-s + 2.23i·7-s + 2.23i·8-s − 9-s − 4·11-s − 1.61i·12-s + 6.47i·13-s − 1.38·14-s + 1.85·16-s − 1.23i·17-s − 0.618i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 0.437i·2-s − 0.577i·3-s + 0.809·4-s + 0.252·6-s + 0.845i·7-s + 0.790i·8-s − 0.333·9-s − 1.20·11-s − 0.467i·12-s + 1.79i·13-s − 0.369·14-s + 0.463·16-s − 0.299i·17-s − 0.145i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.449913921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.449913921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618iT - 2T^{2} \) |
| 7 | \( 1 - 2.23iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6.47iT - 13T^{2} \) |
| 17 | \( 1 + 1.23iT - 17T^{2} \) |
| 23 | \( 1 - 3.23iT - 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 2.76iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 0.472iT - 47T^{2} \) |
| 53 | \( 1 + 5iT - 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 - 7.94T + 61T^{2} \) |
| 67 | \( 1 - 10.9iT - 67T^{2} \) |
| 71 | \( 1 - 9.18T + 71T^{2} \) |
| 73 | \( 1 - 3.47iT - 73T^{2} \) |
| 79 | \( 1 - 2.29T + 79T^{2} \) |
| 83 | \( 1 + 6.94iT - 83T^{2} \) |
| 89 | \( 1 - 9.94T + 89T^{2} \) |
| 97 | \( 1 - 1.70iT - 97T^{2} \) |
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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
−9.600478139029401674881372840849, −8.890010090224651958691210217402, −8.012104231008925313540316728308, −7.23251238433814585648432165782, −6.75607299530934056354079478074, −5.61509771488877472224313916475, −5.29152918455228038983265075880, −3.67232117945070526315595066969, −2.38200944548796501452422769612, −1.86413328042835971710134406324,
0.52101797173706123406503859616, 2.14578158825900728380585635354, 3.20519613158112948527303564123, 3.82045091169646273098333367898, 5.21324918922325643592157226714, 5.74832453796289410688024100113, 7.01848540634882661125118536224, 7.65865809745115121913663766820, 8.367295713229787162850577829261, 9.646148089359530648613594662240
