L(s) = 1 | − 2-s + (0.321 − 1.70i)3-s + 4-s + (2.20 + 0.350i)5-s + (−0.321 + 1.70i)6-s − 8-s + (−2.79 − 1.09i)9-s + (−2.20 − 0.350i)10-s + 3.78i·11-s + (0.321 − 1.70i)12-s − 6.40·13-s + (1.30 − 3.64i)15-s + 16-s + 5.57i·17-s + (2.79 + 1.09i)18-s + 2.10i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.185 − 0.982i)3-s + 0.5·4-s + (0.987 + 0.156i)5-s + (−0.131 + 0.694i)6-s − 0.353·8-s + (−0.930 − 0.365i)9-s + (−0.698 − 0.110i)10-s + 1.14i·11-s + (0.0929 − 0.491i)12-s − 1.77·13-s + (0.337 − 0.941i)15-s + 0.250·16-s + 1.35i·17-s + (0.658 + 0.258i)18-s + 0.482i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.116957627\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.116957627\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.321 + 1.70i)T \) |
| 5 | \( 1 + (-2.20 - 0.350i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 3.78iT - 11T^{2} \) |
| 13 | \( 1 + 6.40T + 13T^{2} \) |
| 17 | \( 1 - 5.57iT - 17T^{2} \) |
| 19 | \( 1 - 2.10iT - 19T^{2} \) |
| 23 | \( 1 - 4.70T + 23T^{2} \) |
| 29 | \( 1 - 7.55iT - 29T^{2} \) |
| 31 | \( 1 - 3.43iT - 31T^{2} \) |
| 37 | \( 1 + 2.75iT - 37T^{2} \) |
| 41 | \( 1 - 8.77T + 41T^{2} \) |
| 43 | \( 1 - 7.04iT - 43T^{2} \) |
| 47 | \( 1 + 2.55iT - 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 1.48iT - 67T^{2} \) |
| 71 | \( 1 - 5.06iT - 71T^{2} \) |
| 73 | \( 1 + 3.33T + 73T^{2} \) |
| 79 | \( 1 + 7.07T + 79T^{2} \) |
| 83 | \( 1 + 9.49iT - 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 4.90T + 97T^{2} \) |
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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
−9.475565381213765153852238321638, −8.937844686064641310753697918983, −7.87243996918321739161090474927, −7.21770895454111972907319258569, −6.64856407104946491596853014098, −5.72905597245956279708228849181, −4.78773485094487735269081800130, −3.05839391551394835895884242249, −2.15776999492825747867296626593, −1.43526241432158350662993620932,
0.53289964946374124859749590162, 2.47364910309595701227727645600, 2.89481089322261295261918472862, 4.48561061293037273196814670718, 5.27412861252796247407739286879, 6.00292856597012971123659527524, 7.11742825982788609363592599319, 7.956278710647792757750622639607, 9.063157895786842928562561277041, 9.311288659107498110510759694525