| 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
−13.67824470380584011116251191708, −11.79392955181948897054147586143, −11.06467434938477418279673629635, −10.11485635869289555045207872539, −8.906409122405316760789388547047, −8.503752662946548951874176437651, −7.19288394612326020712882680933, −4.78667662397843245370527530688, −3.33193189380058538457018703174, −1.31230137948293484711659380068,
1.63611321449537247582520448422, 3.86971093880859423594161990682, 6.24425673840059993360632195380, 7.35541454808651941564686954979, 8.143596866151099796351559559617, 9.063380271929813916318231709529, 9.886926977946388235014032083998, 11.68191553558567183812956150486, 12.44476908275827937354971224887, 14.00720356245934848711272896944
