| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 3.44i·7-s + i·8-s − 9-s + 2·11-s − i·12-s + 6.89i·13-s + 3.44·14-s + 16-s + 0.550i·17-s + i·18-s + 2.89·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.30i·7-s + 0.353i·8-s − 0.333·9-s + 0.603·11-s − 0.288i·12-s + 1.91i·13-s + 0.921·14-s + 0.250·16-s + 0.133i·17-s + 0.235i·18-s + 0.665·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.360537414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.360537414\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 7 | \( 1 - 3.44iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 6.89iT - 13T^{2} \) |
| 17 | \( 1 - 0.550iT - 17T^{2} \) |
| 19 | \( 1 - 2.89T + 19T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 6.34iT - 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 - 1.10iT - 43T^{2} \) |
| 47 | \( 1 + 5.89iT - 47T^{2} \) |
| 53 | \( 1 + 8.89iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 4.89T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 8.79T + 71T^{2} \) |
| 73 | \( 1 - 11.8iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 7.44iT - 83T^{2} \) |
| 89 | \( 1 + 4.34T + 89T^{2} \) |
| 97 | \( 1 + 16.6iT - 97T^{2} \) |
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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
−8.885192749338611693937700659506, −8.602466135286437722467225487832, −7.38252498664513062797299926607, −6.43352479748432628318212783865, −5.71072026713631358578971684874, −4.86092399153171814491843824523, −4.14733239349505450860186938304, −3.31782774610525249852778278394, −2.33320761038587508261393165932, −1.52947805622538728006903128878,
0.44387248881990854465148200192, 1.27112340744080385891752220452, 2.87466449234017416947250732469, 3.71113825117504061842887386645, 4.56783493174812715562605016383, 5.57189115202501328154161489118, 6.10647674254019253939582572882, 7.08116288852729734824894371175, 7.58666933433585379831993303583, 7.961641763417719625296600204478
