| L(s) = 1 | + 3i·7-s + 4·11-s + i·13-s + 4i·17-s + 19-s + 4i·23-s − 4·31-s + 9i·37-s − 8i·43-s + 12i·47-s − 2·49-s − 8i·53-s − 4·59-s − 5·61-s − 11i·67-s + ⋯ |
| L(s) = 1 | + 1.13i·7-s + 1.20·11-s + 0.277i·13-s + 0.970i·17-s + 0.229·19-s + 0.834i·23-s − 0.718·31-s + 1.47i·37-s − 1.21i·43-s + 1.75i·47-s − 0.285·49-s − 1.09i·53-s − 0.520·59-s − 0.640·61-s − 1.34i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.707508734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707508734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 9iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 8iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 5iT - 97T^{2} \) |
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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
−8.470360958805363914906805967017, −7.79069649305554665643431146012, −6.84075251870433879531768741192, −6.23677329590089121039487044327, −5.62242918571371554028641927105, −4.80208071255630176286821099423, −3.87509137148009522121856517899, −3.19696358568573385484819134056, −2.07461257374055932086993191387, −1.34570797303715441464810617132,
0.46952281390110531252636695270, 1.37354839828689688535446946434, 2.56688594706573486144994342202, 3.59748960480907163482422588339, 4.14910827577004919924162215779, 4.92855318238906594259106760175, 5.84210667852036916964007291831, 6.66682891041724654249548330039, 7.21496693114564215195257678505, 7.76541664297207504092583255487
