| L(s) = 1 | + i·3-s + 4.60i·7-s − 9-s + 11-s + 4.60i·13-s + 6.60i·17-s + 7.21·19-s − 4.60·21-s − i·27-s − 8·29-s + 9.21·31-s + i·33-s − 3.21i·37-s − 4.60·39-s + 8·41-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.74i·7-s − 0.333·9-s + 0.301·11-s + 1.27i·13-s + 1.60i·17-s + 1.65·19-s − 1.00·21-s − 0.192i·27-s − 1.48·29-s + 1.65·31-s + 0.174i·33-s − 0.527i·37-s − 0.737·39-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.757843155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.757843155\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 4.60iT - 7T^{2} \) |
| 13 | \( 1 - 4.60iT - 13T^{2} \) |
| 17 | \( 1 - 6.60iT - 17T^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 9.21T + 31T^{2} \) |
| 37 | \( 1 + 3.21iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 3.39iT - 43T^{2} \) |
| 47 | \( 1 + 5.21iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 0.605iT - 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 10.6iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 12.4iT - 97T^{2} \) |
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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
−9.110516294824944537174950174077, −8.397525027167683083507312784412, −7.60047230999296492899763683205, −6.45419328687959756519576983793, −5.89837858271101721020373867234, −5.23542774585475426852524098836, −4.30664234400951404837029041111, −3.47494691814784888647452617164, −2.48274112861992828261075328657, −1.59001085215154776795230136545,
0.60668013558785614254164741314, 1.18514889724109959461487313027, 2.78762871994149745641135505964, 3.43213604584140230553107789600, 4.46828264662698163022500192260, 5.26199752458697766975634672725, 6.14121584611446878084434511575, 7.13311232977488341617051106668, 7.49710023338533356006736102160, 7.946960795514801759008557039950
