| L(s) = 1 | + 2.44i·2-s − 5·3-s + 10·4-s + (−5 + 24.4i)5-s − 12.2i·6-s + (−35 + 34.2i)7-s + 63.6i·8-s − 56·9-s + (−59.9 − 12.2i)10-s + 89·11-s − 50·12-s + 5·13-s + (−84 − 85.7i)14-s + (25 − 122. i)15-s + 4.00·16-s + 485·17-s + ⋯ |
| L(s) = 1 | + 0.612i·2-s − 0.555·3-s + 0.625·4-s + (−0.200 + 0.979i)5-s − 0.340i·6-s + (−0.714 + 0.699i)7-s + 0.995i·8-s − 0.691·9-s + (−0.599 − 0.122i)10-s + 0.735·11-s − 0.347·12-s + 0.0295·13-s + (−0.428 − 0.437i)14-s + (0.111 − 0.544i)15-s + 0.0156·16-s + 1.67·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.839i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.564677 + 1.03737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.564677 + 1.03737i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (5 - 24.4i)T \) |
| 7 | \( 1 + (35 - 34.2i)T \) |
| good | 2 | \( 1 - 2.44iT - 16T^{2} \) |
| 3 | \( 1 + 5T + 81T^{2} \) |
| 11 | \( 1 - 89T + 1.46e4T^{2} \) |
| 13 | \( 1 - 5T + 2.85e4T^{2} \) |
| 17 | \( 1 - 485T + 8.35e4T^{2} \) |
| 19 | \( 1 - 220. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 700. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 191T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.05e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.63e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.91e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 377. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.19e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 1.58e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.62e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.93e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.04e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 4.45e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 8.65e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.56e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.99e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 808. iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.23e3T + 8.85e7T^{2} \) |
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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
−16.32059356744224888236700699846, −14.98014542926527280332458792812, −14.34350143146305992649782016101, −12.18780273573968715198874847638, −11.44070189882575411020251376976, −10.07097693322161259283997131319, −8.134074552846417593436736188004, −6.57354826322717129890171805563, −5.84053442731215097730335420704, −2.94758631634964076960185359270,
0.928285038264232944680980413603, 3.57956700653466011907015963814, 5.70706770608959593601195736919, 7.30324564692594695078584080633, 9.242173355310444402337589436805, 10.57242669315692193649474573994, 11.82903793454783615842268947815, 12.48867412441494126891130975416, 13.95439614707832904826478242169, 15.75967767595344737108854696118
