| L(s) = 1 | + i·3-s + 2.24·5-s + (0.274 − 2.63i)7-s − 9-s − 5.61i·11-s + 3.17i·13-s + 2.24i·15-s − 0.653·17-s + 1.73·19-s + (2.63 + 0.274i)21-s + (3.98 + 2.66i)23-s + 0.0182·25-s − i·27-s + 0.830·29-s − 3.41i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.00·5-s + (0.103 − 0.994i)7-s − 0.333·9-s − 1.69i·11-s + 0.881i·13-s + 0.578i·15-s − 0.158·17-s + 0.398·19-s + (0.574 + 0.0599i)21-s + (0.831 + 0.555i)23-s + 0.00365·25-s − 0.192i·27-s + 0.154·29-s − 0.613i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.025177575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.025177575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (-0.274 + 2.63i)T \) |
| 23 | \( 1 + (-3.98 - 2.66i)T \) |
| good | 5 | \( 1 - 2.24T + 5T^{2} \) |
| 11 | \( 1 + 5.61iT - 11T^{2} \) |
| 13 | \( 1 - 3.17iT - 13T^{2} \) |
| 17 | \( 1 + 0.653T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 29 | \( 1 - 0.830T + 29T^{2} \) |
| 31 | \( 1 + 3.41iT - 31T^{2} \) |
| 37 | \( 1 + 9.03iT - 37T^{2} \) |
| 41 | \( 1 + 4.58iT - 41T^{2} \) |
| 43 | \( 1 - 1.46iT - 43T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 0.509iT - 53T^{2} \) |
| 59 | \( 1 - 1.43iT - 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 4.99iT - 67T^{2} \) |
| 71 | \( 1 + 3.32T + 71T^{2} \) |
| 73 | \( 1 - 10.8iT - 73T^{2} \) |
| 79 | \( 1 + 3.98iT - 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 + 2.63T + 97T^{2} \) |
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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
−9.174398347691969623207069774253, −8.559323905079231355728686072155, −7.51029051287766931522663404972, −6.64997011787163317088609968191, −5.81465947958568425631107590781, −5.18700765695650079155479265987, −4.04056983556216500306640985381, −3.37556960354439587350813559108, −2.11370456601401321204344675839, −0.76881638237726863728162182076,
1.38954727065411299202400362279, 2.28547349474947936351865255295, 3.02182713640077673241992520237, 4.70486900655697135639861472516, 5.27468628017830602085490431773, 6.16458147737745399026180224128, 6.81911649960044501967611591504, 7.71918330795838850788260763564, 8.497297868670536420338666259505, 9.379383335177184891171148941896
