L(s) = 1 | + 1.41i·2-s + 2.29i·3-s − 2.00·4-s + (2.28 − 4.44i)5-s − 3.23·6-s + 9.92·7-s − 2.82i·8-s + 3.75·9-s + (6.29 + 3.22i)10-s + 3.77i·11-s − 4.58i·12-s − 6.77i·13-s + 14.0i·14-s + (10.1 + 5.22i)15-s + 4.00·16-s + 8.36·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.763i·3-s − 0.500·4-s + (0.456 − 0.889i)5-s − 0.539·6-s + 1.41·7-s − 0.353i·8-s + 0.417·9-s + (0.629 + 0.322i)10-s + 0.342i·11-s − 0.381i·12-s − 0.521i·13-s + 1.00i·14-s + (0.679 + 0.348i)15-s + 0.250·16-s + 0.491·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.57846 + 1.04963i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57846 + 1.04963i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 5 | \( 1 + (-2.28 + 4.44i)T \) |
| 23 | \( 1 + (-1.76 - 22.9i)T \) |
good | 3 | \( 1 - 2.29iT - 9T^{2} \) |
| 7 | \( 1 - 9.92T + 49T^{2} \) |
| 11 | \( 1 - 3.77iT - 121T^{2} \) |
| 13 | \( 1 + 6.77iT - 169T^{2} \) |
| 17 | \( 1 - 8.36T + 289T^{2} \) |
| 19 | \( 1 - 3.59iT - 361T^{2} \) |
| 29 | \( 1 - 5.85T + 841T^{2} \) |
| 31 | \( 1 + 42.7T + 961T^{2} \) |
| 37 | \( 1 + 4.28T + 1.36e3T^{2} \) |
| 41 | \( 1 + 25.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 65.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 5.16iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 8.63T + 2.80e3T^{2} \) |
| 59 | \( 1 - 82.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 108. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 30.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 117.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 81.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 138. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 33.5T + 6.88e3T^{2} \) |
| 89 | \( 1 - 49.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 138.T + 9.40e3T^{2} \) |
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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
−12.29454069334549668376307561758, −11.05143493148597912623784183591, −10.00530375493068452133296502110, −9.201160038902052266479412675548, −8.206186954165878498467548721345, −7.34516740547436683460160328642, −5.57946705483053906946738988517, −4.99746079619072496131703910570, −3.99334208539356941714454158245, −1.50514987124522523936604740931,
1.40813370408760870059613383182, 2.46209047115308927343818509114, 4.15628864294948467370450771017, 5.53563654293553003278600670498, 6.87447295358031661374682217953, 7.76997654389223168298876055108, 8.872268588033984917991423362627, 10.15056160241284813748186781871, 10.95879524056788999634469694872, 11.69719009799099415626845582631