| L(s) = 1 | + i·2-s − 4-s + (−2.23 − 0.133i)5-s + i·7-s − i·8-s + (0.133 − 2.23i)10-s + 1.73·11-s − 1.46i·13-s − 14-s + 16-s + 4i·17-s − 2.26·19-s + (2.23 + 0.133i)20-s + 1.73i·22-s + 4.46i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.998 − 0.0599i)5-s + 0.377i·7-s − 0.353i·8-s + (0.0423 − 0.705i)10-s + 0.522·11-s − 0.406i·13-s − 0.267·14-s + 0.250·16-s + 0.970i·17-s − 0.520·19-s + (0.499 + 0.0299i)20-s + 0.369i·22-s + 0.930i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2545403958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2545403958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.23 + 0.133i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 1.46iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 2.26T + 19T^{2} \) |
| 23 | \( 1 - 4.46iT - 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 + 3.73T + 31T^{2} \) |
| 37 | \( 1 + 7.19iT - 37T^{2} \) |
| 41 | \( 1 + 9.92T + 41T^{2} \) |
| 43 | \( 1 + 11.4iT - 43T^{2} \) |
| 47 | \( 1 - 4.53iT - 47T^{2} \) |
| 53 | \( 1 + 13.8iT - 53T^{2} \) |
| 59 | \( 1 + 8.92T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 8.39iT - 67T^{2} \) |
| 71 | \( 1 + 7.73T + 71T^{2} \) |
| 73 | \( 1 - 3.07iT - 73T^{2} \) |
| 79 | \( 1 - 4.92T + 79T^{2} \) |
| 83 | \( 1 + 2.53iT - 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−8.768840289271078816200182115445, −8.239340846817290133473422121277, −7.43341633997287827852265906306, −6.77797780689626554159074045361, −5.79575039009538513048931511681, −5.10052095241760077791358654605, −3.93602354787666963181843024633, −3.49658733988231193144339104905, −1.81591455868524523220047236424, −0.10075589716188714063255672904,
1.26690708558688412324259878211, 2.64600816576134119874240097826, 3.59968655464971679516703056166, 4.33679405458420453504468707838, 5.03853588503454388608697997591, 6.39654970150806372109949012547, 7.12496784214881256565960743055, 7.950728858466433971924103225016, 8.742908432183982245585333109809, 9.410461258826638362335459236996
