| L(s) = 1 | + 2.09i·3-s − 6.85·7-s + 4.62·9-s + (−10.3 − 3.61i)11-s + 8.70·13-s + 30.8·17-s + 13.4i·19-s − 14.3i·21-s + 8.15i·23-s + 28.4i·27-s − 32.9i·29-s − 33.0·31-s + (7.55 − 21.7i)33-s + 58.3i·37-s + 18.2i·39-s + ⋯ |
| L(s) = 1 | + 0.697i·3-s − 0.978·7-s + 0.513·9-s + (−0.944 − 0.328i)11-s + 0.669·13-s + 1.81·17-s + 0.708i·19-s − 0.682i·21-s + 0.354i·23-s + 1.05i·27-s − 1.13i·29-s − 1.06·31-s + (0.228 − 0.658i)33-s + 1.57i·37-s + 0.466i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.191126634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.191126634\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (10.3 + 3.61i)T \) |
| good | 3 | \( 1 - 2.09iT - 9T^{2} \) |
| 7 | \( 1 + 6.85T + 49T^{2} \) |
| 13 | \( 1 - 8.70T + 169T^{2} \) |
| 17 | \( 1 - 30.8T + 289T^{2} \) |
| 19 | \( 1 - 13.4iT - 361T^{2} \) |
| 23 | \( 1 - 8.15iT - 529T^{2} \) |
| 29 | \( 1 + 32.9iT - 841T^{2} \) |
| 31 | \( 1 + 33.0T + 961T^{2} \) |
| 37 | \( 1 - 58.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 80.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 38.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 40.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 0.654iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 111. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 23.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 70.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 56.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 46.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 151.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 137.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 154. iT - 9.40e3T^{2} \) |
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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
−9.825558666489180184803131888433, −9.577492182957307245262892405867, −8.214936791502023097094520748189, −7.67622573589879439887363590865, −6.45302578252126592772814977560, −5.71657386751735990186978678301, −4.78763543782859464024054249633, −3.58920684209177568066181263213, −3.10305143824911604969025443152, −1.32883976501874962910937090092,
0.38189496439613778328913505538, 1.70691544424565063744484808279, 2.96497424588323382934633863422, 3.86904196511234392556970123831, 5.25135694233760948212248250370, 5.97103719656840613665431755650, 7.15964599497626466016748135545, 7.34423295080161422243794162751, 8.501383219918354076640798606466, 9.368586603043675279283491853302
