| L(s) = 1 | − 3-s − 1.84i·5-s + 9-s − 5.22i·11-s + 4.46i·13-s + 1.84i·15-s + 2.93i·17-s + 5.65·19-s − 2.16i·23-s + 1.58·25-s − 27-s − 5.41·29-s + 9.65·31-s + 5.22i·33-s + 1.41·37-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.826i·5-s + 0.333·9-s − 1.57i·11-s + 1.23i·13-s + 0.477i·15-s + 0.710i·17-s + 1.29·19-s − 0.451i·23-s + 0.317·25-s − 0.192·27-s − 1.00·29-s + 1.73·31-s + 0.909i·33-s + 0.232·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.362485306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.362485306\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.84iT - 5T^{2} \) |
| 11 | \( 1 + 5.22iT - 11T^{2} \) |
| 13 | \( 1 - 4.46iT - 13T^{2} \) |
| 17 | \( 1 - 2.93iT - 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 2.16iT - 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 - 9.65T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 + 4.01iT - 41T^{2} \) |
| 43 | \( 1 + 3.06iT - 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 1.39iT - 61T^{2} \) |
| 67 | \( 1 + 13.5iT - 67T^{2} \) |
| 71 | \( 1 - 6.49iT - 71T^{2} \) |
| 73 | \( 1 - 4.90iT - 73T^{2} \) |
| 79 | \( 1 + 11.7iT - 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 + 9.05iT - 89T^{2} \) |
| 97 | \( 1 - 9.23iT - 97T^{2} \) |
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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
−8.770349945480418114780270393974, −8.222467298034588278050966531583, −7.21148410114947122863520457441, −6.30841562512505362466084337100, −5.71520306341466831760459369912, −4.87967084877586384608896834488, −4.08237623411166074745723648960, −3.08725971063358345708231687965, −1.61049140143573995827314163094, −0.60334433333222168708839991967,
1.11456933102686322758731587252, 2.52464494927919617870615438511, 3.29489312991416412464782763490, 4.52431060928792218274667896972, 5.20873921394203099062030772232, 6.03538927915580055116666514504, 7.01521289585263138226514945881, 7.38541742417998755897676217761, 8.167541168741954617564433265368, 9.564643880842570584983830483979
