| L(s) = 1 | − 0.324i·3-s + i·5-s + 2.45·7-s + 2.89·9-s + i·11-s + 5.71i·13-s + 0.324·15-s − 1.52·17-s − 3.45i·19-s − 0.797i·21-s + 5.50·23-s − 25-s − 1.91i·27-s + 7.02i·29-s − 3.73·31-s + ⋯ |
| L(s) = 1 | − 0.187i·3-s + 0.447i·5-s + 0.929·7-s + 0.964·9-s + 0.301i·11-s + 1.58i·13-s + 0.0837·15-s − 0.369·17-s − 0.791i·19-s − 0.174i·21-s + 1.14·23-s − 0.200·25-s − 0.368i·27-s + 1.30i·29-s − 0.670·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.026484366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.026484366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 11 | \( 1 - iT \) |
| good | 3 | \( 1 + 0.324iT - 3T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 13 | \( 1 - 5.71iT - 13T^{2} \) |
| 17 | \( 1 + 1.52T + 17T^{2} \) |
| 19 | \( 1 + 3.45iT - 19T^{2} \) |
| 23 | \( 1 - 5.50T + 23T^{2} \) |
| 29 | \( 1 - 7.02iT - 29T^{2} \) |
| 31 | \( 1 + 3.73T + 31T^{2} \) |
| 37 | \( 1 - 5.29iT - 37T^{2} \) |
| 41 | \( 1 + 5.86T + 41T^{2} \) |
| 43 | \( 1 + 2.69iT - 43T^{2} \) |
| 47 | \( 1 - 1.30T + 47T^{2} \) |
| 53 | \( 1 + 1.45iT - 53T^{2} \) |
| 59 | \( 1 - 6.53iT - 59T^{2} \) |
| 61 | \( 1 + 4.33iT - 61T^{2} \) |
| 67 | \( 1 + 0.782iT - 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 9.17iT - 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 - 4.15T + 97T^{2} \) |
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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
−9.267127820085861442553247731425, −8.754466617113363397546831004071, −7.65079953521708392927857225041, −6.93018576094559233835311830273, −6.59755875384478867475800666396, −5.05395681939418092800995091627, −4.61001430933277040289666656670, −3.55985951749071661612430956930, −2.19505015411815775710307342329, −1.40571636707014826678673610637,
0.842346239110800612713286353173, 1.97668667553322942607623766255, 3.33508599203642990439675951050, 4.28484026395001822398771298852, 5.11130901987704287234587959775, 5.74422969307403194398982679376, 6.91154962728391533017890608373, 7.85478427698503319196495475083, 8.216999732960689695864508761722, 9.217149709916056284614484399506
