L(s) = 1 | + (−2.31 + 1.33i)2-s + (2.46 − 1.71i)3-s + (1.58 − 2.74i)4-s + (−1.93 + 1.11i)5-s + (−3.40 + 7.27i)6-s + (4.89 + 5.00i)7-s − 2.23i·8-s + (3.10 − 8.44i)9-s + (2.99 − 5.18i)10-s + (15.6 + 9.04i)11-s + (−0.812 − 9.46i)12-s + 2.70·13-s + (−18.0 − 5.05i)14-s + (−2.84 + 6.07i)15-s + (9.32 + 16.1i)16-s + (−3.39 − 1.96i)17-s + ⋯ |
L(s) = 1 | + (−1.15 + 0.669i)2-s + (0.820 − 0.572i)3-s + (0.395 − 0.685i)4-s + (−0.387 + 0.223i)5-s + (−0.567 + 1.21i)6-s + (0.699 + 0.714i)7-s − 0.279i·8-s + (0.345 − 0.938i)9-s + (0.299 − 0.518i)10-s + (1.42 + 0.822i)11-s + (−0.0677 − 0.788i)12-s + 0.208·13-s + (−1.28 − 0.360i)14-s + (−0.189 + 0.405i)15-s + (0.582 + 1.00i)16-s + (−0.199 − 0.115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.942421 + 0.396519i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.942421 + 0.396519i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.46 + 1.71i)T \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
| 7 | \( 1 + (-4.89 - 5.00i)T \) |
good | 2 | \( 1 + (2.31 - 1.33i)T + (2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (-15.6 - 9.04i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 2.70T + 169T^{2} \) |
| 17 | \( 1 + (3.39 + 1.96i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-14.7 - 25.6i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (5.66 - 3.27i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 18.8iT - 841T^{2} \) |
| 31 | \( 1 + (-12.6 + 21.9i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (33.5 + 58.0i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 38.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 63.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-33.6 + 19.4i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (0.787 + 0.454i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (20.6 + 11.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (25.3 + 43.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (34.7 - 60.1i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 55.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-14.8 + 25.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (14.9 + 25.9i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 78.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (133. - 77.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 32.8T + 9.40e3T^{2} \) |
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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
−14.00720356245934848711272896944, −12.44476908275827937354971224887, −11.68191553558567183812956150486, −9.886926977946388235014032083998, −9.063380271929813916318231709529, −8.143596866151099796351559559617, −7.35541454808651941564686954979, −6.24425673840059993360632195380, −3.86971093880859423594161990682, −1.63611321449537247582520448422,
1.31230137948293484711659380068, 3.33193189380058538457018703174, 4.78667662397843245370527530688, 7.19288394612326020712882680933, 8.503752662946548951874176437651, 8.906409122405316760789388547047, 10.11485635869289555045207872539, 11.06467434938477418279673629635, 11.79392955181948897054147586143, 13.67824470380584011116251191708