L(s) = 1 | + (−1.33 + 1.49i)2-s + 1.73i·3-s + (−0.446 − 3.97i)4-s + (−2.58 − 2.30i)6-s + 6.56i·7-s + (6.52 + 4.63i)8-s − 2.99·9-s + 2.26i·11-s + (6.88 − 0.773i)12-s + 14.8·13-s + (−9.79 − 8.75i)14-s + (−15.6 + 3.55i)16-s − 26.8·17-s + (3.99 − 4.47i)18-s + 10.8i·19-s + ⋯ |
L(s) = 1 | + (−0.666 + 0.745i)2-s + 0.577i·3-s + (−0.111 − 0.993i)4-s + (−0.430 − 0.384i)6-s + 0.938i·7-s + (0.815 + 0.579i)8-s − 0.333·9-s + 0.206i·11-s + (0.573 − 0.0644i)12-s + 1.14·13-s + (−0.699 − 0.625i)14-s + (−0.975 + 0.221i)16-s − 1.57·17-s + (0.222 − 0.248i)18-s + 0.572i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0404866 - 0.722732i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0404866 - 0.722732i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.33 - 1.49i)T \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 6.56iT - 49T^{2} \) |
| 11 | \( 1 - 2.26iT - 121T^{2} \) |
| 13 | \( 1 - 14.8T + 169T^{2} \) |
| 17 | \( 1 + 26.8T + 289T^{2} \) |
| 19 | \( 1 - 10.8iT - 361T^{2} \) |
| 23 | \( 1 - 36.4iT - 529T^{2} \) |
| 29 | \( 1 + 35.2T + 841T^{2} \) |
| 31 | \( 1 + 23.8iT - 961T^{2} \) |
| 37 | \( 1 + 54.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 23.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 56.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 51.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 30.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 6.92iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 107.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 111. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 31.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 110.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 59.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 142. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.14T + 7.92e3T^{2} \) |
| 97 | \( 1 - 126.T + 9.40e3T^{2} \) |
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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
−11.61085217108177735751814324838, −10.97734602849784213710387235453, −9.883845676049147115156031588832, −9.004483633631729718210186223461, −8.486902083507835990114532599374, −7.20916590621286021636555373939, −6.02681951678452036736400738917, −5.31399808206552118680714492159, −3.88597423827901572785661086495, −1.93265448734993787335902132294,
0.43153919398526176434424569808, 1.88800396186055897657657732647, 3.39324805148440310685448830527, 4.56033601452852005560190214867, 6.49161344877504850789967673117, 7.20084006441738401872945863104, 8.440151561673237726134785241852, 8.952647627420599031114660904641, 10.39450929433013063244620668635, 10.94688728082115450968078848070