| L(s) = 1 | + (2 − 2i)3-s + 5i·5-s + (2 + 2i)7-s + i·9-s + 8·11-s + (−3 + 3i)13-s + (10 + 10i)15-s + (7 + 7i)17-s + 20i·19-s + 8·21-s + (−2 + 2i)23-s − 25·25-s + (20 + 20i)27-s − 40i·29-s + 52·31-s + ⋯ |
| L(s) = 1 | + (0.666 − 0.666i)3-s + i·5-s + (0.285 + 0.285i)7-s + 0.111i·9-s + 0.727·11-s + (−0.230 + 0.230i)13-s + (0.666 + 0.666i)15-s + (0.411 + 0.411i)17-s + 1.05i·19-s + 0.380·21-s + (−0.0869 + 0.0869i)23-s − 25-s + (0.740 + 0.740i)27-s − 1.37i·29-s + 1.67·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.04539 + 0.581055i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04539 + 0.581055i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5iT \) |
| good | 3 | \( 1 + (-2 + 2i)T - 9iT^{2} \) |
| 7 | \( 1 + (-2 - 2i)T + 49iT^{2} \) |
| 11 | \( 1 - 8T + 121T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 169iT^{2} \) |
| 17 | \( 1 + (-7 - 7i)T + 289iT^{2} \) |
| 19 | \( 1 - 20iT - 361T^{2} \) |
| 23 | \( 1 + (2 - 2i)T - 529iT^{2} \) |
| 29 | \( 1 + 40iT - 841T^{2} \) |
| 31 | \( 1 - 52T + 961T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-42 + 42i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (18 + 18i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (53 - 53i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 20iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 48T + 3.72e3T^{2} \) |
| 67 | \( 1 + (62 + 62i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 28T + 5.04e3T^{2} \) |
| 73 | \( 1 + (47 - 47i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + (18 - 18i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 80iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (63 + 63i)T + 9.40e3iT^{2} \) |
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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
−11.64607882151072088403009525243, −10.51892705650010151059998770332, −9.680039127101922974825941665738, −8.398460943915667755131748714658, −7.74916034382571295725625967703, −6.76985651513538834443181265468, −5.81786934139138164078574115614, −4.12905377665079554164938187557, −2.83173947653165560421207226790, −1.73673106186176142693196532384,
1.05555449912427093422493478929, 2.97789672215722418063483479817, 4.24787538765960950245126687354, 4.98333481203219928433741495413, 6.44606412800501350919894234585, 7.75152450861019584063470430515, 8.739644392277153183017969501821, 9.328045504588263865534143261686, 10.14479492757461918283613385651, 11.39327676336287180249698022478
