| L(s) = 1 | − 1.41i·2-s + 1.73·3-s − 2.00·4-s − 2.44i·6-s + (6.63 + 2.24i)7-s + 2.82i·8-s + 2.99·9-s − 10.2·11-s − 3.46·12-s − 8.95·13-s + (3.17 − 9.37i)14-s + 4.00·16-s − 30.4·17-s − 4.24i·18-s − 16.1i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.500·4-s − 0.408i·6-s + (0.947 + 0.320i)7-s + 0.353i·8-s + 0.333·9-s − 0.931·11-s − 0.288·12-s − 0.689·13-s + (0.226 − 0.669i)14-s + 0.250·16-s − 1.78·17-s − 0.235i·18-s − 0.849i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.043805081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.043805081\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-6.63 - 2.24i)T \) |
| good | 11 | \( 1 + 10.2T + 121T^{2} \) |
| 13 | \( 1 + 8.95T + 169T^{2} \) |
| 17 | \( 1 + 30.4T + 289T^{2} \) |
| 19 | \( 1 + 16.1iT - 361T^{2} \) |
| 23 | \( 1 + 6.72iT - 529T^{2} \) |
| 29 | \( 1 + 30T + 841T^{2} \) |
| 31 | \( 1 + 50.1iT - 961T^{2} \) |
| 37 | \( 1 + 30.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 7.10iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 74.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 58.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 70.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 0.492iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 2.86iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 27.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 50.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 70.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 133.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 104.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 144. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 100.T + 9.40e3T^{2} \) |
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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
−9.204104270990287157043386414659, −8.717555662282132615521546242908, −7.80661989931934904394265859584, −7.06466288989372708529775328900, −5.60666837318099908176916493970, −4.75393712074217288594007587042, −3.97672078498655144393589980261, −2.41792520358083221819932736173, −2.17929225163690731501125354646, −0.27618175968791770979463784608,
1.62143940326523710664262219940, 2.83306561183549535993028376963, 4.22977052652093758561460144039, 4.84817898372594019968804889881, 5.84199917894686106142460832005, 7.05948649386764507853770894561, 7.58148761798671287165562491013, 8.396865976976764231431741464448, 8.998533814762311998893049708625, 10.04669891549042893098982061721
