| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 0.449i·7-s − i·8-s − 9-s − 1.44·11-s + i·12-s − 2.44i·13-s + 0.449·14-s + 16-s − 0.449i·17-s − i·18-s − 19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.169i·7-s − 0.353i·8-s − 0.333·9-s − 0.437·11-s + 0.288i·12-s − 0.679i·13-s + 0.120·14-s + 0.250·16-s − 0.109i·17-s − 0.235i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4546500723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4546500723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 0.449iT - 7T^{2} \) |
| 11 | \( 1 + 1.44T + 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 0.449iT - 17T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 11.7iT - 37T^{2} \) |
| 41 | \( 1 - 8.89T + 41T^{2} \) |
| 43 | \( 1 + 2.44iT - 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 2.55iT - 53T^{2} \) |
| 59 | \( 1 + 1.55T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 + 9.24iT - 67T^{2} \) |
| 71 | \( 1 + 6.44T + 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 6.34iT - 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 6.44iT - 97T^{2} \) |
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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
−9.025931856374606076010116511329, −7.995226266359165281671293403438, −7.69028658225655078392644225928, −6.93010631912321600420630260639, −6.03435238698647121610543937394, −5.49891449964781878440156524082, −4.57575483495006454252791013390, −3.55317430740943283979481505112, −2.55224890405005306703709103283, −1.23786343270479931839241356768,
0.14911792150051679509744781023, 1.82329461313093143762950952066, 2.63292388368798581690813628711, 3.78769250621256491878452272675, 4.23361953428399078707473720747, 5.35420925729400302286250178947, 5.83111810978684199698158270201, 7.06989814372131366780016858156, 7.79656768039263447581259709297, 8.854222666723128597594198348012
