L(s) = 1 | + (2.99 − 0.164i)3-s + (−3.61 + 3.61i)5-s − 12.2i·7-s + (8.94 − 0.985i)9-s + (1.76 − 1.76i)11-s + (2.38 − 2.38i)13-s + (−10.2 + 11.4i)15-s − 20.0i·17-s + (8.77 − 8.77i)19-s + (−2.02 − 36.7i)21-s + 13.1·23-s − 1.10i·25-s + (26.6 − 4.42i)27-s + (−6.51 − 6.51i)29-s + 37.5·31-s + ⋯ |
L(s) = 1 | + (0.998 − 0.0548i)3-s + (−0.722 + 0.722i)5-s − 1.75i·7-s + (0.993 − 0.109i)9-s + (0.160 − 0.160i)11-s + (0.183 − 0.183i)13-s + (−0.681 + 0.761i)15-s − 1.18i·17-s + (0.461 − 0.461i)19-s + (−0.0962 − 1.75i)21-s + 0.573·23-s − 0.0443i·25-s + (0.986 − 0.163i)27-s + (−0.224 − 0.224i)29-s + 1.21·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.88916 - 0.943535i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.88916 - 0.943535i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.99 + 0.164i)T \) |
good | 5 | \( 1 + (3.61 - 3.61i)T - 25iT^{2} \) |
| 7 | \( 1 + 12.2iT - 49T^{2} \) |
| 11 | \( 1 + (-1.76 + 1.76i)T - 121iT^{2} \) |
| 13 | \( 1 + (-2.38 + 2.38i)T - 169iT^{2} \) |
| 17 | \( 1 + 20.0iT - 289T^{2} \) |
| 19 | \( 1 + (-8.77 + 8.77i)T - 361iT^{2} \) |
| 23 | \( 1 - 13.1T + 529T^{2} \) |
| 29 | \( 1 + (6.51 + 6.51i)T + 841iT^{2} \) |
| 31 | \( 1 - 37.5T + 961T^{2} \) |
| 37 | \( 1 + (10.0 + 10.0i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 4.57T + 1.68e3T^{2} \) |
| 43 | \( 1 + (21.2 + 21.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 54.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-21.5 + 21.5i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (53.6 - 53.6i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (-19.2 + 19.2i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (31.5 - 31.5i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 65.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 50.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 20.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + (6.35 + 6.35i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 166.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 139.T + 9.40e3T^{2} \) |
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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
−10.89276277879173885414742466626, −10.11216877238414957550410616846, −9.181188640910610021602947524853, −7.935225107236442233732740632416, −7.32263505765960904228868700016, −6.74083198280941000705162876178, −4.69318319876303503625278640601, −3.72324168828394867337606448788, −2.91982561975315305839089490374, −0.915345119695454790440431874652,
1.68366108747882099320568970592, 2.99382382499328499270667705706, 4.18113575914085733098177179253, 5.29366628758689054024631615938, 6.53805342535160153993757132491, 7.983912052646465561711771540751, 8.493528526692682759879603130827, 9.132630379034599988307117179440, 10.09487866785733417648113770765, 11.49454050308068839196205151765