| L(s) = 1 | + 5i·3-s − 8.66i·5-s − 17.3·7-s + 2·9-s + 11i·11-s − 34.6i·13-s + 43.3·15-s + 30·17-s + 44i·19-s − 86.6i·21-s + 60.6·23-s + 49.9·25-s + 145i·27-s − 34.6i·29-s − 337.·31-s + ⋯ |
| L(s) = 1 | + 0.962i·3-s − 0.774i·5-s − 0.935·7-s + 0.0740·9-s + 0.301i·11-s − 0.739i·13-s + 0.745·15-s + 0.428·17-s + 0.531i·19-s − 0.899i·21-s + 0.549·23-s + 0.399·25-s + 1.03i·27-s − 0.221i·29-s − 1.95·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.058067141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.058067141\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 11iT \) |
| good | 3 | \( 1 - 5iT - 27T^{2} \) |
| 5 | \( 1 + 8.66iT - 125T^{2} \) |
| 7 | \( 1 + 17.3T + 343T^{2} \) |
| 13 | \( 1 + 34.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 30T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 60.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 34.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 337.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 303. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 72T + 6.89e4T^{2} \) |
| 43 | \( 1 - 250iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 121.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 138. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 471iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 415. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 205iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 233.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 520T + 3.89e5T^{2} \) |
| 79 | \( 1 - 883.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 210iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 849T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−10.18304706484572327687277404881, −9.576271739547080449143707230777, −8.921145927158841001659440310853, −7.87035066624277682179003729592, −6.83395688297768145900927018170, −5.65308493704922534879232641960, −4.90253169063688627414418419899, −3.90204189001728850028313549060, −3.02311197703921426135885601894, −1.23833535800072167094986644350,
0.31149104993959568823642689041, 1.77420113392262687064626350589, 2.91096887249461627264166473513, 3.89419525178114323547815431117, 5.42059909026763693157850725701, 6.53639979431196604597675945524, 6.94148436389894175269334630604, 7.67905673206160159683779951345, 8.944766571736113300664228499834, 9.639808871182585247200424321279
