L(s) = 1 | + (−1.95 − 0.438i)2-s + 1.73·3-s + (3.61 + 1.71i)4-s + (−3.37 − 0.758i)6-s − 6.33·7-s + (−6.30 − 4.92i)8-s + 2.99·9-s + 9.27i·11-s + (6.26 + 2.96i)12-s + 18.5i·13-s + (12.3 + 2.77i)14-s + (10.1 + 12.3i)16-s − 13.9i·17-s + (−5.85 − 1.31i)18-s + 17.2i·19-s + ⋯ |
L(s) = 1 | + (−0.975 − 0.219i)2-s + 0.577·3-s + (0.904 + 0.427i)4-s + (−0.563 − 0.126i)6-s − 0.904·7-s + (−0.788 − 0.615i)8-s + 0.333·9-s + 0.843i·11-s + (0.521 + 0.246i)12-s + 1.42i·13-s + (0.882 + 0.198i)14-s + (0.634 + 0.772i)16-s − 0.818i·17-s + (−0.325 − 0.0730i)18-s + 0.907i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0218 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0218 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.581499 + 0.594371i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.581499 + 0.594371i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.95 + 0.438i)T \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 6.33T + 49T^{2} \) |
| 11 | \( 1 - 9.27iT - 121T^{2} \) |
| 13 | \( 1 - 18.5iT - 169T^{2} \) |
| 17 | \( 1 + 13.9iT - 289T^{2} \) |
| 19 | \( 1 - 17.2iT - 361T^{2} \) |
| 23 | \( 1 + 33.7T + 529T^{2} \) |
| 29 | \( 1 - 28.6T + 841T^{2} \) |
| 31 | \( 1 - 23.4iT - 961T^{2} \) |
| 37 | \( 1 - 67.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 50.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 31.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 81.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 19.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 53.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.49T + 4.48e3T^{2} \) |
| 71 | \( 1 - 13.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 141. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 69.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 46.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + 68.5iT - 9.40e3T^{2} \) |
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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
−11.96625991167361401094566106853, −10.43101304267244055697602033269, −9.758484108314535543596693555316, −9.118098135074972095803551160485, −8.067439194830436651972481234843, −7.05159832162769833148412211187, −6.30654625649469566787785056357, −4.29776640809274763660449023949, −2.99493642534246269967281113579, −1.72679125243687387077968248699,
0.49912882759719528029302456955, 2.46768337788018379203128083792, 3.58653258337369377926184318181, 5.64947707276177460625775277844, 6.48421220925715521436422110294, 7.71762244516436385271033825525, 8.407287686773347420665588364181, 9.333840435813151599568543154213, 10.21991831666753379369732871852, 10.88523166277178538396268508193