| L(s) = 1 | − i·3-s − 9-s + 5·11-s + 5i·13-s + i·17-s − 19-s + 3i·23-s + i·27-s − 2·29-s + 2·31-s − 5i·33-s − 8i·37-s + 5·39-s − 5·41-s + 9i·43-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.333·9-s + 1.50·11-s + 1.38i·13-s + 0.242i·17-s − 0.229·19-s + 0.625i·23-s + 0.192i·27-s − 0.371·29-s + 0.359·31-s − 0.870i·33-s − 1.31i·37-s + 0.800·39-s − 0.780·41-s + 1.37i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.637075813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637075813\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 - iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−8.413782709613363425400511060934, −7.44249739653859149819360093415, −6.95942140907161037551967712069, −6.25155590236636979879636803239, −5.70878801861162108616857857153, −4.42735981545243608358314581413, −4.03357718576532469689459991069, −2.95814679677296858328679923491, −1.83865951614441373932293222772, −1.22921252021196055227534814782,
0.45564920569513548924692311771, 1.67725627908677512592235431669, 2.90731481366993754623906963499, 3.57865854563388281536423591791, 4.36804294936162604396060142269, 5.13194698337087848091571785031, 5.91766934071070267201344842979, 6.60176686326953904337528581102, 7.35642174088452516940996931971, 8.297850737510501390494400127690
