L(s) = 1 | + (−0.534 + 1.92i)2-s + 1.73i·3-s + (−3.42 − 2.05i)4-s + (−3.33 − 0.925i)6-s − 11.9i·7-s + (5.79 − 5.51i)8-s − 2.99·9-s + 14.5i·11-s + (3.56 − 5.94i)12-s + 22.4·13-s + (23.0 + 6.39i)14-s + (7.52 + 14.1i)16-s + 12.6·17-s + (1.60 − 5.78i)18-s − 8.76i·19-s + ⋯ |
L(s) = 1 | + (−0.267 + 0.963i)2-s + 0.577i·3-s + (−0.857 − 0.514i)4-s + (−0.556 − 0.154i)6-s − 1.71i·7-s + (0.724 − 0.688i)8-s − 0.333·9-s + 1.32i·11-s + (0.297 − 0.495i)12-s + 1.72·13-s + (1.64 + 0.456i)14-s + (0.470 + 0.882i)16-s + 0.746·17-s + (0.0890 − 0.321i)18-s − 0.461i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.19626 + 0.677142i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19626 + 0.677142i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.534 - 1.92i)T \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 11.9iT - 49T^{2} \) |
| 11 | \( 1 - 14.5iT - 121T^{2} \) |
| 13 | \( 1 - 22.4T + 169T^{2} \) |
| 17 | \( 1 - 12.6T + 289T^{2} \) |
| 19 | \( 1 + 8.76iT - 361T^{2} \) |
| 23 | \( 1 - 4.99iT - 529T^{2} \) |
| 29 | \( 1 - 2.74T + 841T^{2} \) |
| 31 | \( 1 + 16.3iT - 961T^{2} \) |
| 37 | \( 1 - 32.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 42.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 16.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 48.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 94.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 43.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 56.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 61.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 39.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 99.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 10.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 140. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 54.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 14.1T + 9.40e3T^{2} \) |
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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
−11.36249908469139262964568163019, −10.36952473203679723048397597725, −9.886099956260114833027360722458, −8.767697113425463068902351931279, −7.67189101078367121170007584804, −6.96677448602879162102702632442, −5.80381633022608374044439941485, −4.45353987570142024503241733034, −3.83307896495019608215957845310, −1.02052608948419132526273470163,
1.14132465030501480292308923314, 2.60029944476819899145037096747, 3.61796906999894973479839425140, 5.49683947177391183730078399492, 6.14726298091961183278218554708, 8.058113165538913876855276918135, 8.582170690195476535305569755477, 9.309122130735938949319098211786, 10.72233279609414034000887144979, 11.44459855663201574636547984777