| L(s) = 1 | + (0.271 − 1.01i)2-s + (0.777 + 0.449i)4-s + (−0.310 + 2.21i)5-s + (−1.26 + 2.32i)7-s + (2.15 − 2.15i)8-s + (2.16 + 0.915i)10-s + (−1.52 − 0.880i)11-s + (4.98 + 4.98i)13-s + (2.00 + 1.91i)14-s + (−0.698 − 1.20i)16-s + (0.458 − 0.122i)17-s + (2.33 − 1.34i)19-s + (−1.23 + 1.58i)20-s + (−1.30 + 1.30i)22-s + (−6.30 − 1.69i)23-s + ⋯ |
| L(s) = 1 | + (0.192 − 0.716i)2-s + (0.388 + 0.224i)4-s + (−0.138 + 0.990i)5-s + (−0.479 + 0.877i)7-s + (0.760 − 0.760i)8-s + (0.683 + 0.289i)10-s + (−0.460 − 0.265i)11-s + (1.38 + 1.38i)13-s + (0.537 + 0.512i)14-s + (−0.174 − 0.302i)16-s + (0.111 − 0.0298i)17-s + (0.535 − 0.309i)19-s + (−0.276 + 0.354i)20-s + (−0.278 + 0.278i)22-s + (−1.31 − 0.352i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57969 + 0.176914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57969 + 0.176914i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.310 - 2.21i)T \) |
| 7 | \( 1 + (1.26 - 2.32i)T \) |
| good | 2 | \( 1 + (-0.271 + 1.01i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (1.52 + 0.880i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.98 - 4.98i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.458 + 0.122i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.33 + 1.34i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.30 + 1.69i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 7.01T + 29T^{2} \) |
| 31 | \( 1 + (-3.95 + 6.84i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.45 + 1.73i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 6.20iT - 41T^{2} \) |
| 43 | \( 1 + (2.28 + 2.28i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.393 - 1.47i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.26 - 8.45i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (6.17 - 10.7i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.26 + 3.91i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.530 + 1.98i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 9.79iT - 71T^{2} \) |
| 73 | \( 1 + (1.45 - 0.391i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-9.08 + 5.24i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.07 - 2.07i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.64 - 2.85i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.21 + 3.21i)T - 97iT^{2} \) |
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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
−11.80596039447486639895475898022, −10.88051819467080072603307367210, −10.15795850305528390079727311120, −8.999239067779847924113248319607, −7.85107246697862720193928535692, −6.66632728482469619186563195690, −6.03342208055061291006645275815, −4.12498455215363641931023928575, −3.11624630890626801842783842473, −2.09978065020531732209136057919,
1.23970888360901935907712918238, 3.40097586181919338833783581487, 4.79592171170903722118959883312, 5.73857599854728911474070555866, 6.67211299310288696271067212692, 7.925583209088179338147844091245, 8.320049141380480400144845768333, 9.988232719943967282633743480105, 10.49544957676672583728200411369, 11.67932262113555821871673900200