| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s − 4·11-s + i·12-s + 6i·13-s + 14-s + 16-s − 2i·17-s + i·18-s + 4·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 1.20·11-s + 0.288i·12-s + 1.66i·13-s + 0.267·14-s + 0.250·16-s − 0.485i·17-s + 0.235i·18-s + 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9973740073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9973740073\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.878281140281778810296220485956, −9.307215598219349727443615852565, −8.381398093182565916602536198385, −7.54081455174277370117470627374, −6.69789449947503591538753446986, −5.51543246987784864238787750689, −4.80612288257266238837571564354, −3.46547048821906674206182966468, −2.48067351646546667041921647201, −1.43589860649440578340056872560,
0.46322738231351764285810569445, 2.69699799632198537109020192183, 3.68933291315861357170597846532, 4.88720046406530784640123264312, 5.43742236789017511637270918504, 6.38405499306493241796379451237, 7.55424875193128866131440014876, 8.078940799036683463191045508709, 8.858629048484985253080819050319, 10.08988142246171816333105146519
