L(s) = 1 | + (−2.63 − 2.63i)2-s + (2.54 − 1.59i)3-s + 9.88i·4-s + (4.15 − 2.78i)5-s + (−10.8 − 2.49i)6-s + (6.57 − 2.39i)7-s + (15.5 − 15.5i)8-s + (3.91 − 8.10i)9-s + (−18.2 − 3.60i)10-s + 5.66i·11-s + (15.7 + 25.1i)12-s + (1.68 − 1.68i)13-s + (−23.6 − 11.0i)14-s + (6.11 − 13.6i)15-s − 42.1·16-s + (−14.0 + 14.0i)17-s + ⋯ |
L(s) = 1 | + (−1.31 − 1.31i)2-s + (0.846 − 0.531i)3-s + 2.47i·4-s + (0.830 − 0.556i)5-s + (−1.81 − 0.415i)6-s + (0.939 − 0.342i)7-s + (1.93 − 1.93i)8-s + (0.434 − 0.900i)9-s + (−1.82 − 0.360i)10-s + 0.514i·11-s + (1.31 + 2.09i)12-s + (0.129 − 0.129i)13-s + (−1.68 − 0.786i)14-s + (0.407 − 0.913i)15-s − 2.63·16-s + (−0.824 + 0.824i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.499105 - 0.973920i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.499105 - 0.973920i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.54 + 1.59i)T \) |
| 5 | \( 1 + (-4.15 + 2.78i)T \) |
| 7 | \( 1 + (-6.57 + 2.39i)T \) |
good | 2 | \( 1 + (2.63 + 2.63i)T + 4iT^{2} \) |
| 11 | \( 1 - 5.66iT - 121T^{2} \) |
| 13 | \( 1 + (-1.68 + 1.68i)T - 169iT^{2} \) |
| 17 | \( 1 + (14.0 - 14.0i)T - 289iT^{2} \) |
| 19 | \( 1 + 24.0T + 361T^{2} \) |
| 23 | \( 1 + (-3.17 + 3.17i)T - 529iT^{2} \) |
| 29 | \( 1 - 24.1T + 841T^{2} \) |
| 31 | \( 1 - 23.8iT - 961T^{2} \) |
| 37 | \( 1 + (13.8 - 13.8i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 53.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-25.9 - 25.9i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-27.3 + 27.3i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-22.4 + 22.4i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 14.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 90.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (0.492 - 0.492i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 54.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (30.4 - 30.4i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 58.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-55.7 - 55.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 109. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (48.0 + 48.0i)T + 9.40e3iT^{2} \) |
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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
−12.88555963106463566662353724491, −12.11743834968850010372587331118, −10.73831246737070948765528216829, −9.952910081242455146823159700469, −8.624878410617906739733158792165, −8.430747053504656033453472252998, −6.91496021488073450172941253373, −4.24555712214746688556035672941, −2.34533341253338967305835472565, −1.37930725559908685957762691505,
2.10776973351888749159279285859, 4.93691713575373807543307270669, 6.25770314225606644060699300454, 7.47342262177575161526164011349, 8.608014949617149719098449316034, 9.151798465376365811469023803120, 10.32278618605971614564968790576, 11.05617221654177103774632795451, 13.60970004707241726731879077380, 14.28141891957975957728458713213